Quantum criticality of spinons

TL;DR

This paper rigorously analyzes the quantum criticality of spinons in 1D antiferromagnetic spin-1/2 chains, revealing how spin strings influence critical properties and identifying distinct quantum phases through exact solutions.

Contribution

It provides an exact analytical study of the quantum critical behavior of spinons using Bethe ansatz, highlighting the role of spin strings and phase crossover temperatures.

Findings

Double peaks in specific heat mark two crossover temperatures.

Wilson ratio remains nearly temperature-independent in TLL phase.

Precise quantum scalings and critical exponents for magnetic properties are determined.

Abstract

The free fermion nature of interacting spins in one dimensional (1D) spin chains still lacks a rigorous study. In this letter we show that the length-$1$ spin strings significantly dominate critical properties of spinons, magnons and free fermions in the 1D antiferromagnetic spin-1/2 chain. Using the Bethe ansatz solution we analytically calculate exact scaling functions of thermal and magnetic properties of the model, providing a rigorous understanding of the quantum criticality of spinons. It turns out that the double peaks in specific heat elegantly mark two crossover temperatures fanning out 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 $R_W=4K_s$ remains almost temperature-independent, here $K_s$ is the Luttinger parameter. Furthermore, applying our results we precisely determine the quantum scalings and critical exponents of all magnetic properties in the ideal 1D spin-1/2 antiferromagnet Cu(C${}_4$H${}_4$N${}_2$)(NO${}_3$)${}_2$ recently studied in Phys. Rev. Lett. {\bf 114}, 037202 (2015)]. We further find that the magnetization peak used in experiments is not a good quantity to map out the finite temperature TLL phase boundary.

Equations

$$\mathcal{H}=2J\sum _{j=1}^N{\vec{S}}_j\cdot {\vec{S}}_{j+1}-g{\mu }_{B}H{M}^z,$$

$${\lambda }_{j,\ell }^n={\lambda }_j^n+\frac{1}{2}\mathrm{i}(n+1-2\ell )$$

$${\epsilon }_n^{+}={\epsilon }_n^0-\sum _m{A}_{m,n}\ast {\epsilon }_m^{-}(\lambda ),$$

$$R_{\mathrm{W}}=4K_s$$

$$f$$

$$M^z$$

$$c_v$$

$$\alpha =\beta =1/2,\delta =2,z=2,\nu =\frac{1}{2}.$$

$$\chi =\frac{1}{2\sqrt{\pi JT}}{e}^{-{\Delta }_g/T}$$

$${\left(\frac{{\lambda }_j-\frac{i}{2}}{{\lambda }_j+\frac{i}{2}}\right)}^2=-\prod _{l=1}^M\frac{{\lambda }_j-{\lambda }_l-i}{{\lambda }_j-{\lambda }_l+i},$$

$$E({\lambda }_1,\cdots ,{\lambda }_M)=-\sum _{j=1}^M\left(\frac{J}{{\lambda }_j^2+\frac{1}{4}}\right)+HM+E_0.$$

$${\lambda }_{j,l}^n={\lambda }_j^n+\frac{i}{2}(n+1-2l)$$

$${\epsilon }_n^{+}={\epsilon }_n^0-\sum _m{A}_{m,n}\ast {\epsilon }_n^{-}$$

$$A_{m,n}\left(\lambda \right)=a_{m+n}\left(\lambda \right)+2a_{m+n-2}\left(\lambda \right)+\cdots +2a_{|m-n|+2}\left(\lambda \right)+a_{|m-n|}.$$

$$f=\sum _n{\int }_{-\infty }^{\infty }{a}_n\left(\lambda \right){\epsilon }_n^{-}\left(\lambda \right)d\lambda .$$

$${\epsilon }_1^{\left(0\right)}\left(\lambda \right)$$

$$f_0$$

$${\epsilon }_1^{(0)}(\lambda )\approx -2\pi J{a}_1(\lambda )+H-\frac{1}{\pi }{\int }_{-Q}^Q{\epsilon }_1^{(0)}(\lambda )\mathrm{d}\lambda \approx -2\pi J{a}_1(\lambda )+H-\frac{2Q(H-H_s)}{\pi }.$$

$$f_0$$

$$M^z$$

$$\chi$$

$$1-\frac{M^z}{M_s}=D{\left(1-\frac{H}{H_s}\right)}^{1/\delta }=\frac{4}{\pi }{\left(1-\frac{H}{H_s}\right)}^{1/2}$$

$${\rho }_1(\lambda )+{\rho }_1^{\mathrm{h}}(\lambda )=a_1(\lambda )-(a_2\ast {\rho }_1)(\lambda )-\left((a_{\nu -1}+a_{\nu +1})\ast {\rho }_{\nu }\right)(\lambda ).$$

$$\vartheta =2(\gamma -v_{\ell }).$$

$${\epsilon }_1\left(\lambda \right)$$

$$\eta \left(\lambda \right)$$

$${\epsilon }_1\left(\lambda \right)=t\left(\lambda -Q\right)+O\left({\left(\lambda -Q\right)}^2\right)$$

$$I$$

$$\eta \left(\lambda \right)=\frac{{\pi }^2{T}^2}{6t}\left[a_2\left(\lambda +Q\right)+a_2\left(\lambda -Q\right)\right]-{\int }_{-Q}^Q{a}_2\left(\lambda -\mu \right)\eta \left(\mu \right)d\mu .$$

$$f_0\left(0,H\right)={\int }_{-Q}^Q{a}_1\left(\lambda \right){\epsilon }_1^{\left(0\right)}\left(\lambda \right)d\lambda .$$

$$f\left(T,H\right)=-T{\int }_{-\infty }^{\infty }{a}_1\left(\lambda \right)\ln \left(1+e^{\frac{-{\epsilon }_{{}_1}\left(\lambda \right)}{T}}.\right)d\lambda .$$

Full text

Of central importance to the study of the 1D spin-1/2 antiferromagnetic Heisenberg chain is the understanding of spin excitations Yang:1966a ; Faddeev:1981 ; Haldane:1981 ; Affleck:1986a ; Takahashi:1999 ; WangYP:2015 ; Johnston:2000 ; Tennant:1995 ; Lake:2005 ; Mourigal:435 ; Lake:2013 ; Zheludev:2008 ; Stone:2003 . Elementary spin excitations in this model may exhibit quasi-particle behaviour which is described by spinons carrying half a unit of spin. Such fractional quasiparticles are responsible for the TLL in the model Affleck:1986 ; Cardy:1986 ; Giamarchi:2004 .

Regarding to the Bethe ansatz solution of the 1D spin-1/2 chain, a significant development is Takahashi’s discovery of spin string patterns Takahashi:1971 , i.e., magnon bound states with different string lengths. Takahashi’s spin strings give one full access to the thermodynamics of the model through Yang and Yang’s grand canonical approach Yang:1969 , namely the so-called thermodynamic Bethe ansatz (TBA) equations Takahashi:1971 . However, the problems of how such spin strings determine the free fermion nature of spinons and how spin strings comprise universal scalings of thermal and magnetic properties still lack a rigorous understanding. In this paper we present a full answer to these questions.

Using spin string solutions to the TBA equations, we obtain the following results: I) we obtain exact scaling functions, critical exponents and a benchmark of quantum magnetism for the 1D spin-1/2 Heisenberg chain, revealing the microscopic origin of the quasiparticle spinons, free fermions and magnons that emerge in different physical regimes; II) We find that the Wilson ratio Som28 ; Wil75 , the ratio between the susceptibility $\chi$ and the specific heat $c_{v}$ divided by the temperature $T$, $R_{\mathrm{W}}=\frac{4}{3}\left(\frac{\pi k_{B}}{g\mu_{B}}\right)^{2}\chi/(c_{v}/T)$, significantly characterises the TLL of spinons and marks the crossover temperature between the quantum critical phase and the TLL Kono:2015 , see Fig. 1. When the magnetic field is larger than the saturation field, dilute magnon behaviour is evidenced by the exponential decay of the susceptibility; III) Using our analytical and numerical results we precisely determine the quantum scalings and magnetic properties of the ideal spin-1/2 antiferromagnet Cu(C4H4N2)(NO3)2 (denoted by CuPzN for short) Kono:2015 . We also find that the magnetization peak used in experiment Kono:2015 ; Ruegg:2008 ; Shaginyan:2016 is not a good quantity to map out the finite temperature TLL phase boundary. Instead one should use the Wilson ratio or the specific heat peaks.

Bethe ansatz equations. The Hamiltonian of the 1D Heisenberg spin 1/2 chain is given by Bethe:1931

[TABLE]

where $J$ is the intrachain coupling constant, $N$ is the number of lattice sites and $M^{z}=\sum_{j=1}^{N}S_{j}^{z}=N/2-M$ is the magnetization. $M$ is the number of down spins. In this Hamiltonian, $g$ and $\mu_{B}$ are the Landé factor and the Bohr magneton, respectively. To simplify notation, we let $g\mu_{B}=1$. The spin-1/2 operator $\vec{S}_{j}$ associate to the site $j$ interacts with its nearest neighbours under a magnetic field $H$. The energy is given by $E=-\sum^{M}_{j=1}\frac{J}{\lambda_{j}^{2}+\frac{1}{4}}+HM+E_{0}$, where $E_{0}=\frac{1}{2}N\left(J-H\right)$, and the spin quasimomenta $\lambda_{j}$ with $j=1,\ldots,M$ are determined by the Bethe ansatz (BA) equations Bethe:1931 ; Takahashi:1999 , also see Supp . For the ground state, all the $\lambda_{j}$ take real values. However, at finite temperatures and in the thermodynamic limit, there are real and complex solutions describing different lengths of bound states

[TABLE]

with $\ell=1,\ldots,n$, and $j=1,\ldots,\nu_{n}$. Here $\lambda^{n}_{j}$ and $\nu_{n}$ denote the real part and the number of length-$n$ strings, respectively Faddeev:1981 .

Building on such spin strings Takahashi:1971 , the thermodynamics of the system is determined by the TBA equations

[TABLE]

where $\ast$ denotes convolution, $n$ takes positive integer values and $\varepsilon_{n}^{\pm}=\pm T\ln[1+{\rm e}^{\pm\varepsilon_{n}/T}]$ defines the dressed energy of the length-$n$ spin strings. The driving term is given by $\varepsilon_{n}^{0}=-2\pi Ja_{n}(\lambda)+nH$ with the kernel function $a_{n}(\lambda)=\frac{1}{2\pi}\frac{n}{\lambda^{2}+n^{2}/4}$. The function $A_{m,n}$ is given in Supp . The free energy per unit length is given by $f=\sum_{n}\int a_{n}\left(\lambda\right)\varepsilon_{n}^{-}\left(\lambda\right)d\lambda$. Hereafter, all magnetic properties will be in the per unit lengths.

Spin strings and spin liquid. For low-lying excitations, each magnon decomposes into two spinons, i.e. spin-1/2 quasiparticles Faddeev:1981 ; Hammar:1999 ; Karbach:2000 ; Karbach:2002 ; Caux:2005 ; Caux:2006 ; Klauser:2012 ; YangW:2017 . The spectral weight of two spinon excitations have been experimentally confirmed through observation of the spin dynamic structure factor Lake:2005 ; Mourigal:435 ; Lake:2013 ; Zheludev:2008 ; Stone:2003 . In order to calculate the spin string contributions to the thermodynamics at different temperature scales, we rewrite the free energy as $f=\sum_{n}g_{n}(\lambda)+\sum_{n}\varepsilon_{n}^{-}(\infty)$, where $g_{n}=\int{\rm d\lambda}\>a_{n}(\lambda)(\varepsilon_{n}^{-}(\lambda)-\varepsilon_{n}^{-}(\infty))$ counts the major contribution from the length-$n$ strings, besides their constant values $\varepsilon_{n}^{\pm}(\infty)$, to the free energy. Thus $g_{n}$ is very convenient for estimating the cut-off string length $n_{c}$, see Supp . It is important to observe that $g_{n}$ shows a power law decay as $n$ increases, see Fig. 1(b).

Here we observe that for a small value of $H/T$, a large cut-off string length $n_{c}$ is needed in the calculation of the thermodynamics. When $T\to\infty$, full string patterns are required, i.e. $n_{c}\to\infty$, so that the free energy reduces to that of free spins: $f=\sum_{n}\varepsilon_{n}^{-}(\infty)$. Moreover, for $H\sim 0^{+}$ and $T\ll 1$, logarithmic temperature corrections to the thermodynamical properties of the renormalization fixed point effective Hamiltonian have been seen Johnston:2000 ; Lukyanov:1998 ; Eggert:1994 . At $T=0$, all the $\lambda_{j}$ take real values. In this case, one easily gets the known magnetization critical exponent $\delta=2$ in the scaling form $1-M^{z}/M_{s}=D(1-H/H_{s})^{1/\delta}$ with $D=4/\pi$ Supp . This gives a divergent spin susceptibility at the saturation point $H_{s}=4J$ Bonner:1964 .

At low temperatures, i.e. $T\ll H$, the TLL feature is dominated by the excitations close to the Fermi points of the length-$1$ string $\varepsilon_{1}$ in the parameter $\lambda$ space. Such elementary excitations are described by particle hole excitations. From the TBA equations (S.4), the dressed energy $\varepsilon_{1}$ is given by $\varepsilon_{1}(\lambda)=\varepsilon_{1}^{(0)}(\lambda)+\eta(\lambda)+O(T^{3})$, where $\varepsilon_{1}^{(0)}(\lambda)$ is given by the dressed energy equation (S.4) in the limit $T=0$ and the leading order temperature correction is determined by $\eta(\lambda)=\frac{\pi^{2}T^{2}}{6t}[a_{2}(\lambda+Q)+a_{2}(\lambda-Q)]-\int_{-Q}^{Q}a_{2}\ast\eta(\mu){\rm d}\mu$. Here, $Q$ is fixed by the external field through $\varepsilon_{1}^{\left(0\right)}\left(\pm Q\right)=0$, see Supp . At low temperatures and in the limit of zero magnetic field, the free energy has been calculated by the Wiener-Hopf method Nepomechie:1993 . For arbitrary $H<H_{s}$, we thus obtain the field theory result for the free energy: $f=E_{0}-\pi T^{2}/(6v_{s})+O(T^{3})$, where $E_{0}$ is the ground state energy and the sound velocity is given by $v_{s}=\frac{1}{2\pi}\frac{d\varepsilon_{1}\left(\lambda\right)/d\lambda}{\rho_{0}\left(\lambda\right)}\mid_{\lambda=Q}$ Supp . This free energy gives the relativistic behavour of phonons Affleck:1986a , where the specific heat is $c_{v}/T=\pi/(3v_{s})$. This gives the dynamic critical exponent $z=1$.

Quantum criticality of spinons. In this spin-1/2 chain, the phase transition between the magnetized and ferromagnetic phases occurs at the saturation point Haldane:1981 ; Affleck:1986a ; Maeda:2007 . However, the determination of the phase boundary of the TLL at quantum criticality is still in question. In experiments Ruegg:2008 ; Kono:2015 , the magnetization peaks were regarded as the, as yet unjustified, TLL phase boundary. In Fig. 1 (a), we demonstrate that the peak positions of the specific heat( the dotted solid lines) fanning out from the saturation field $H_{s}$ coincide with the phase boundaries determined by the Wilson ratio $R_{\rm W}$. We observe that the phase boundary of the TLL determined by the magnetization peaks deviates significantly from the true TLL phase boundary as determined by the Wilson ratio and specific heat.

In Fig. 1 (a), we further demonstrate the existence of crossover temperatures from the double-peak structure of the specific heat. The existence of these crossover temperatures results in three different fluctuation regions: quantum and thermal fluctuations reach an equal footing (TLL); thermal fluctuations strongly coupled to quantum fluctuations (QC); dilute magnons dominate the fluctuations (FM). We show that there exists an intrinsic connection between the Wilson ratio and Luttinger parameter

[TABLE]

for the Luttinger liquid, i.e. $H\leq H_{s}$, see Fig. 1(b). Here $K_{s}$ is the Luttinger parameter. A similar relation was recently found in spin ladder compounds and Fermi gases Nin12 ; GuaYFB13 ; Yu:2016 ; Saghafi:2016 . Thus the Wilson ratio elegantly quantifies the TLL regardless of the microscopic details of the underlying quantum system. This elegant relation (4) is confirmed by the numerical solutions of the TBA equations (S.4), see Fig. 1 (b). Moreover, the relation between the Luttinger parameter $K_{s}$ and the sound velocity $K_{s}=\pi v_{s}\chi/\left(g\mu_{B}\right)^{2}$ is also universal Giamarchi:2004 .

We further show that the length-$1$ spin strings dominate the quantum criticality of the antiferromagnetic spin-1/2 chain in the vicinity of the critical point Supp . We prove that the vanishing Fermi point gives rise to a universality class of free fermion criticality, i.e. the dilute spinons. By developing the generating function of free fermions in the TBA equations (S.4) Supp , we obtain the free energy

[TABLE]

near $H_{s}$, where $b_{1}=-\frac{\sqrt{\pi}T^{\frac{3}{2}}}{4\sqrt{J}}\operatorname{Li}_{\frac{3}{2}}\left(-e^{\frac{A}{T}}\right)$ and $b_{2}=-\frac{1}{2}\frac{\sqrt{\pi}T^{\frac{5}{2}}}{\left(16J\right)^{\frac{3}{2}}}\operatorname{Li}_{\frac{5}{2}}\left(-e^{\frac{A}{T}}\right)$ with $A=4J-H-\frac{b_{1}}{\pi}+\frac{b_{2}}{\pi}$. This simple result gives very accurate thermal and magnetic properties for the field near the saturation field, see 1(a). The polylog function $\operatorname{Li}_{3/2}(x)$ appearing in $b_{1}$ indicates that the spinons are similar in nature to free fermions. The magnon density $n_{\rm magnon}=M_{s}/N-M^{z}=\frac{\sqrt{2m^{*}T}}{\pi}\int^{\infty}_{0}\frac{dx}{e^{x^{2}-\frac{H_{s}-H}{T}}+1}$ can be obtained from (5) in the vicinity of the critical point. Here the effective mass of the magnon is given by $m^{*}\approx\frac{1}{2J}\left(1-\frac{T^{1/2}}{\sqrt{\pi J}}\int^{\infty}_{0}\frac{dx}{e^{x^{2}-\frac{H_{s}-H}{T}}+1}\right)$. We observe that the effective mass decreases as the magnetic field moves away from the critical point.

Using the standard thermodynamic relations one can obtain entire scaling functions for the per unit length magnetization and the susceptibility for the region beyond the TLL, i.e. $T\gg H_{s}-H$:

[TABLE]

where $\lambda_{0}=1/(2\sqrt{\pi J})$ and $f_{n}^{s}=\operatorname{Li}_{n}\left(-e^{\frac{\Delta}{T}}\right)$ with $\Delta=4J-H$. These analytical scaling functions signify the free fermion nature of the spinons and correspond to a dynamical critical exponent $z=2$ and a correlation length exponent $\nu=1/2$. In particular, the magnetization $(M_{s}/N-M^{z})/H\propto T^{\beta}$ determines the exponent $\beta=1/2$ in the critical region. The scaling function of the specific heat in the critical regime is given by

[TABLE]

We see that $c_{v}/T\propto T^{-\alpha}$ with $\alpha=1/2$. By definition, the Wilson ratio in the critical region satisfies the scaling behaviour $R_{W}\approx\left(\frac{4\pi k_{B}}{3g\mu_{B}}\right)^{2}f_{-1/2}^{s}/f_{3/2}^{s}$ as $H\to H_{s}$. It follows that the Wilson ratio curves at low temperatures intersect, where the slopes are proportional to $1/T$, see the inset of Fig. 1 (b).

So far, we have analytically obtained all critical exponents in the critical region:

[TABLE]

They satisfy the relation $\alpha+\beta\left(1+\delta\right)=2$. In addition, when the magnetic field slightly exceeds the critical field $H_{s}$, the ferromagnetic ordering leads to a gapped phase where the susceptibility decays exponentially, illustrating the universal behaviour of the dilute magnons

[TABLE]

with $\Delta_{g}=4J-H$, see Fig. 3(a).

Application to the spin material. The analytical results obtained here for the quantum scaling functions (6)–(9) provide a precise understanding of the quantum criticality of the ideal spin-1/2 antiferromagnet CuPzN Kono:2015 , on which high precision measurements of the thermal magnetic properties have been made. Here the best fit of magnetic properties determines the coupling constant $2J=10.81$K, Lande factor $g=2.3$ and the saturation field $H_{s}=13.9941$(T) which only slightly differ from the experimental values $2J=10.8(1)$K, $g=2.3(1)$ and $H_{s}=13.97(6)$(T), respectively. Fig. 3(a) shows excellent agreement between our theoretical results for the susceptibility and the experimental data for the spin-1/2 antiferromagnet CuPzN in the measured region. In particular, one can identify dilute magnon behaviour for magnetic fields exceeding $H_{s}$, see the inset of Fig. 3(a). Indeed, the scaling forms of the susceptibility (6) and specific heat (7) fit quite well with the experimental data, see Fig. 3 (b) and (c). However, we mention a small discrepancy between the theoretical result and experimental data for the susceptibility in a narrow window around the critical point. This is due to a 3D coupling effect, which has also been noted in spin ladder compounds Klanjsek:2008 ; Thielemann:2009 ; Ruegg:2008 .

In Fig. 4 (a), (b), we have compared our theoretical calculations with experimental measurements for the magnetization of the antiferromagnet CuPzN subjected to both weak and strong magnetic fields. There was no theoretical examination on the magnetization data measured in this experiment Kono:2015 . Although there is overall agreement between our results and the data, an obvious discrepancy between theory and experiment was observed for $H\sim J^{{}^{\prime}}$ or $H_{s}-H\sim J^{{}^{\prime}}$ due to 3D interchain coupling. For this model $J^{{}^{\prime}}\approx 0.046$K, see the magnetization curves at $H=14.0$, $13.9$, $13.8$T in Fig. 4 (b). In addition, by properly choosing the cut-off string $n_{c}$, we can analyse the full thermodynamics of the model in the entire temperature regime by solving the TBA equation (S.4). In Fig. 4(c), for the specific heat, $n_{c}=30$ was used.

In summary, we have analytically obtained scaling functions and all the critical exponents of the thermal and magnetic properties of the spin-1/2 chain. This provides a rigorous theoretical understanding of the quantum criticality of spinons that has been observed in the antiferromagnet CuPzN Kono:2015 . We have found that the specific heat peaks elegantly mark the phase boundaries between the different phases at quantum criticality and that the Wilson ratio essentially quantifies the TLL and characterises phase transition regardless of the microscopic details of the systems. Our results also shed light on quantum liquids and the criticality of spinons in a variety of systems of interacting bosons and fermions with internal spin degrees of freedom.

Acknowledgments. The authors thank T. Giamarchi and H. Pu for helpful discussions. This work is supported by the NSFC under grant numbers 11374331 and the key NSFC grant No. 11534014. H.Q.L. acknowledges financial support from NSAF U1530401 and computational resources from the Beijing Computational Science Research Centre.