Excitation Spectrum Gap and Spin-Wave Stiffness of XXZ Heisenberg Chains: Global Renormalization-Group Calculation

TL;DR

This study uses renormalization-group theory to analyze the XXZ Heisenberg chain, providing a comprehensive description of its excitation spectrum, spin-wave velocity, and quantum correlations across all temperatures and anisotropies.

Contribution

It offers the first global, quantitative analysis of the XXZ chain's thermodynamics and excitation spectrum, revealing quantum effects and crossover behaviors.

Findings

Antiferromagnetic spin-liquid spin-wave velocity obtained for all anisotropies.

The Ising-like ferromagnetic excitation spectrum gap identified.

Distinct temperature and anisotropy dependence of specific heat peaks in ferromagnetic and antiferromagnetic chains.

Abstract

The anisotropic XXZ spin-1/2 Heisenberg chain is studied using renormalization-group theory. The specific heats and nearest-neighbor spin-spin correlations are calculated thoughout the entire temperature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Isinglike ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found: The in-plane interaction s_i^x s_j^x + s_i^y s_j^y induces an antiferromagnetic correlation in the out-of-plane s_i^z component, at higher temperatures in the antiferromagnetic XXZ chain, dominantly at low temperatures in the ferromagnetic XXZ chain, and, in-between, at all temperatures in the XY chain. We find that the converse effect also occurs in the antiferromagnetic XXZ chain: an antiferromagnetic s_i^z s_j^z interaction induces a correlation in the s_i^xy component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or XY) anisotropy; (ii) in complete contrast, in the ferromagnet, the value and temperature of the specific heat peak increase with either type of anisotropy.

Figures

Tables

Table 1. Table I: The two-site basis eigenstates that appear in Eq.( 8 ). These are the well-known singlet and triplet states. The state | ϕ 3 ⟩ ket subscript italic-ϕ 3 |\phi_{3}\rangle is obtained by spin reversal from | ϕ 1 ⟩ ket subscript italic-ϕ 1 |\phi_{1}\rangle , with the same eigenvalue.

$p$	$s$	$m_s$	Two-site basis eigenstates
$+$	$1$	$1$	$|{\varphi }_1⟩=|\uparrow \uparrow ⟩$
		$0$	$|{\varphi }_2⟩=\frac{1}{\sqrt{2}}\left\{\right|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩\}$
$-$	$0$	$0$	$|{\varphi }_4⟩=\frac{1}{\sqrt{2}}\left\{\right|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩\}$

Table 2. Table II: The three-site basis eigenstates that appear in Eq.( 8 ) with coefficients σ = ( − J z + 8 ​ J x ​ y 2 + J z 2 ) / 2 ​ J x ​ y , τ = ( J z + 8 ​ J x ​ y 2 + J z 2 ) / 2 ​ J x ​ y formulae-sequence 𝜎 subscript 𝐽 𝑧 8 superscript subscript 𝐽 𝑥 𝑦 2 superscript subscript 𝐽 𝑧 2 2 subscript 𝐽 𝑥 𝑦 𝜏 subscript 𝐽 𝑧 8 superscript subscript 𝐽 𝑥 𝑦 2 superscript subscript 𝐽 𝑧 2 2 subscript 𝐽 𝑥 𝑦 \sigma=(-J_{z}+\sqrt{8J_{xy}^{2}+J_{z}^{2}})/2J_{xy},\tau=(J_{z}+\sqrt{8J_{xy}^{2}+J_{z}^{2}})/2J_{xy} and normalization factors μ 𝜇 \mu , ν 𝜈 \nu . The states | ψ 5 − 8 ⟩ ket subscript 𝜓 5 8 |\psi_{5-8}\rangle are obtained by spin reversal from | ψ 1 − 4 ⟩ ket subscript 𝜓 1 4 |\psi_{1-4}\rangle , with the same respective eigenvalues.

$p$	$m_s$	Three-site basis eigenstates
$+$	$3/2$	$|{\psi }_1⟩=|\uparrow \uparrow \uparrow ⟩$
	$1/2$	$|{\psi }_2⟩=\mu \left\{\right|\uparrow \uparrow \downarrow ⟩+\sigma |\uparrow \downarrow \uparrow ⟩+|\downarrow \uparrow \uparrow ⟩\}$
		$|{\psi }_3⟩=\nu \{-|\uparrow \uparrow \downarrow ⟩+\tau |\uparrow \downarrow \uparrow ⟩-|\downarrow \uparrow \uparrow ⟩\}$
$-$	$1/2$	$|{\psi }_4⟩=\frac{1}{\sqrt{2}}\left\{\right|\uparrow \uparrow \downarrow ⟩-|\downarrow \uparrow \uparrow ⟩\}$

Table 3. Table III: Zero-temperature nearest-neighbor correlations of the spin- 1 2 1 2 \frac{1}{2} XY chain.

Zero-temperature correlations of the spin-$\frac{1}{2}$ XY chain	Exact values from Ref. Lieb	Our RGT results
$⟨s_i^{xy}{s}_j^{xy}⟩$	$0.15915$	$0.17678$
$⟨s_i^z{s}_j^z⟩$	$-0.10132$	$-0.12500$

Equations

$$-\beta \mathcal{H}=\sum _{⟨ij⟩}\left\{\left[J_{xy}\left(s_i^x{s}_j^x+s_i^y{s}_j^y\right)+J_z{s}_i^z{s}_j^z\right]+G\right\},$$

$$-\beta \mathcal{H}=\sum _i\left\{-\beta \mathcal{H}(i,i+1)\right\}.$$

$$\text{mbox}{Tr}_{\text{mbox}odd}{e}^{-\beta \mathcal{H}}=$$

$$=$$

$$\simeq$$

$$=\prod _i^{\text{mbox}odd}{e}^{-{\beta }^{\prime }{\mathcal{H}}^{\prime }(i-1,i+1)}$$

$$e^{-{\beta }^{\prime }{\mathcal{H}}^{\prime }(i,k)}=\text{mbox}{Tr}_j{e}^{\left\{-\beta \mathcal{H}(i,j)-\beta \mathcal{H}(j,k)\right\}},$$

$$\begin{gathered} ⟨u_i{v}_k|e^{-{\beta }^{\prime }{\mathcal{H}}^{\prime }(i,k)}|{\bar{u}}_i{\bar{v}}_k⟩= \\ \sum _{w_j}⟨u_i{w}_j{v}_k|e^{-\beta \mathcal{H}(i,j)-\beta \mathcal{H}(j,k)}|{\bar{u}}_i{w}_j{\bar{v}}_k⟩, \end{gathered}$$

$${\Lambda }_1=$$

$${\Lambda }_4=-\frac{1}{2}{J}_{zxy}^{\prime }-\frac{1}{4}{J}_z^{\prime }+G^{\prime }.$$

$${\lambda }_1$$

$${\lambda }_2$$

$${\lambda }_3$$

$$\begin{gathered} {\gamma }_p\equiv ⟨{\varphi }_p|e^{-{\beta }^{\prime }{\mathcal{H}}^{\prime }(i,k)}|{\varphi }_p⟩=\sum _{\begin{array}{c}u,v,\bar{u},\\ \bar{v},w,q\end{array}}⟨{\varphi }_p|u_i{v}_k⟩⟨u_i{w}_j{v}_k|{\psi }_q⟩\cdot \\ ⟨{\psi }_q|e^{-\beta \mathcal{H}(i,j)-\beta \mathcal{H}(j,k)}|{\psi }_q⟩⟨{\psi }_q|{\bar{u}}_i{w}_j{\bar{v}}_k⟩⟨{\bar{u}}_i{\bar{v}}_k|{\varphi }_p⟩. \end{gathered}$$

$${\gamma }_1=e^{\frac{1}{4}{J}_z^{\prime }+G^{\prime }}=e^{2G-\frac{1}{4}{J}_z}$$

$$\displaystyle-\frac{J_{\!z}\sinh\left(\frac{1}{4}\sqrt{8J_{\!xy}^{2}+J_{\!z}^{2}}\right)}{\sqrt{8J_{\!xy}^{2}+J_{\!z}^{2}}}\Bigg{]}\>,$$

$${\gamma }_2=e^{\frac{1}{2}{J}_{xy}^{\prime }-\frac{1}{4}{J}_z^{\prime }+G^{\prime }}=$$

$$\displaystyle+\frac{J_{\!z}\sinh\left(\frac{1}{4}\sqrt{8J_{\!xy}^{2}+J_{\!z}^{2}}\right)}{\sqrt{8J_{\!xy}^{2}+J_{\!z}^{2}}}\Bigg{]}\>,$$

$${\gamma }_4=e^{-\frac{1}{2}{J}_{xy}^{\prime }-\frac{1}{4}{J}_z^{\prime }+G^{\prime }}$$

$$J_{xy}^{\prime }=\ln \left(\frac{{\gamma }_2}{{\gamma }_4}\right),J_z^{\prime }=\ln \left(\frac{{\gamma }_1^2}{{\gamma }_2{\gamma }_4}\right),G^{\prime }=\frac{1}{4}\ln \left({\gamma }_1^2{\gamma }_2{\gamma }_4\right).$$

$$\begin{split}\hat{T}=\left(\begin{array}[]{ccc}\frac{\partial G^{\prime}}{\partial G}&\frac{\partial G^{\prime}}{\partial J_{\!xy}}&\frac{\partial G^{\prime}}{\partial J_{\!z}}\\ 0&\frac{\partial J_{\!xy}^{\prime}}{\partial J_{\!xy}}&\frac{\partial J_{\!xy}^{\prime}}{\partial J_{\!z}}\\ 0&\frac{\partial J_{\!z}^{\prime}}{\partial J_{\!xy}}&\frac{\partial J_{\!xy}^{\prime}}{\partial J_{\!z}}\\ \end{array}\right)\>,\qquad\qquad\quad\\ \end{split}$$

$$\vec{M}=\left(\begin{array}[]{ccc}1&2\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle&\left\langle s_{i}^{z}s_{j}^{z}\right\rangle\\ \end{array}\right)\>.\qquad\qquad$$

$$M_{\alpha }=\frac{1}{N_{\alpha }}\frac{\partial \ln Z}{\partial K_{\alpha }},$$

$$M_{\alpha }=b^{-d}\sum _{\beta }{M}_{\beta }^{\prime }{T}_{\beta \alpha },T_{\beta \alpha }\equiv \frac{N_{\beta }}{N_{\alpha }}\frac{\partial K_{\beta }^{\prime }}{\partial K_{\alpha }}.$$

$$b^d{\vec{M}}^{\ast }={\vec{M}}^{\ast }\cdot \hat{T},$$

$${\vec{M}}^{(0)}=b^{-nd}{\vec{M}}^{(n)}{\hat{T}}^{(n)}{\hat{T}}^{(n-1)}\cdots {\hat{T}}^{(1)},$$

$$\begin{gathered} C=J_{xy}^2\frac{\partial \left(2⟨s_i^{xy}{s}_j^{xy}⟩+R⟨s_i^z{s}_j^z⟩\right)}{\partial J_{xy}},\text{for }T=1/J_{xy}, \\ C=J_z^2\frac{\partial \left(2⟨s_i^{xy}{s}_j^{xy}⟩/R+⟨s_i^z{s}_j^z⟩\right)}{\partial |J_z|},\text{for }T=1/|J_z|. \end{gathered}$$

$$C=\frac{2+R^2}{16}{J}_{xy}^2-\frac{3R}{32}{J}_{xy}^3+\frac{6-8R^2-R^4}{256}{J}_{xy}^4.$$

Full text

PACS numbers: 67.40.Db, 75.10.Pq, 64.60.Cn, 05.10.Cc

I Introduction

The quantum Heisenberg chain, including the possibility of spin-space anisotropy, is the simplest nontrivial quantum spin system and has thus been widely studied since the very beginning of the spin concept in quantum mechanics Bloch ; Bethe ; Hulthen . Interest in this model continued Lieb ; Katsura ; BF ; Inawashiro ; Yang ; Johnsonb ; Johnson and redoubled with the exposition of its richly varied low-temperature behavior Haldane1980 ; Haldane1982 ; Haldane1983 and of its relevance to high-temperature superconductivity Bednorz ; Manousakis ; Keimer ; Greven ; Birge . It has become clear that antiferromagnetism and superconductivity are firmly related to each other, adjoining and overlapping each other.

A large variety of theoretical tools have been employed in the study of the various isotropic and anisotropic regimes of the quantum Heisenberg chain, including finite-systems extrapolation BF ; Narayanan , linked-cluster Inawashiro and dimer-cluster Karbach expansions, quantum decimation Xi-Yao , decoupled Green’s functions Zhang , quantum transfer matrix Fabricius ; Klumper , high-temperature series expansion Rojas , and numerical evaluation of multiple integrals Bortz . The XXZ Heisenberg chain retains high current interest as a theoretical model Damerau ; Hu with direct experimental relevance Thanos .

In the present paper, a position-space renormalization-group method introduced by Suzuki and Takano Suzuki ; Takano for $d=2$ dimensions and already applied to a number of $d>1$ systems Suzuki ; Takano ; Falicov ; Tomczak ; TomRich1 ; TomRich2 ; Hinczewski is used to compute the spin-spin correlations and the specific heat of the $d=1$ anisotropic quantum XXZ Heisenberg model, easily resulting in a global description and detailed quantitative information for the entire temperature and anisotropy ranges including the ferromagnetic and antiferromagnetic, the spin-liquid and Isinglike regions. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Isinglike ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of other characteristics of purely quantum nature are found: The in-plane interaction $s_{i}^{x}s_{j}^{x}+s_{i}^{y}s_{j}^{y}$ induces an antiferromagnetic correlation in the out-of-plane $s_{i}^{z}$ component, at higher temperatures in the antiferromagnetic XXZ chain, dominantly at low temperatures in the ferromagnetic XXZ chain, and, in-between, at all temperatures in the XY chain. We find that the converse effect also occurs in the antiferromagnetic XXZ chain: an antiferromagnetic $s_{i}^{z}s_{j}^{z}$ interaction induces a correlation in the $s_{i}^{xy}$ component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or XY) anisotropy; (ii) in complete contrast, in the ferromagnet, the value of the specific heat peak is strongly dependent on anisotropy and the temperature of the specific heat peak increases with either type of anisotropy. This purely quantum effect is a precursor to different phase transition temperatures in three dimensions Rushbrooke ; Oitmaa ; Falicov ; Hinczewski . Our calculational method is relatively simple, readily yields global results, and is overall quantitatively successful.

II The Anisotropic Quantum Heisenberg Model and the Renormalization-Group Method

II.1 The Anisotropic Quantum Heisenberg Model

The spin-$\frac{1}{2}$ anisotropic Heisenberg model (XXZ model) is defined by the dimensionless Hamiltonian

[TABLE]

where $\beta=1/k_{B}T$ and $\langle ij\rangle$ denotes summation over nearest-neighbor pairs of sites. Here the $s_{i}^{u}$ are the quantum mechanical Pauli spin operators at site $i$. The additive constant $G$ is generated by the renormalization-group transformation and is used in the calculation of thermodynamic functions. The anisotropy coefficient is $R=J_{z}/J_{xy}$. The model reduces to the isotropic Heisenberg model (XXX model) for $|R|=1$, to the XY model for $R=0$, and to the Ising model for $|R|\rightarrow\infty$.

II.2 Renormalization-Group Recursion Relations

The Hamiltonian in Eq.(1) can be rewritten as

[TABLE]

where $\beta\mathcal{H}(i,i+1)$ is a Hamiltonian involving sites $i$ and $i+1$ only. The renormalization-group procedure, which eliminates half of the degrees of freedom and keeps the partition function unchanged, is done approximately Suzuki ; Takano :

[TABLE]

Here and throughout this paper, the primes are used for the renormalized system. Thus, at each successive length scale, we ignore the non-commutativity of the operators beyond three consecutive sites, in the two steps indicated by $\simeq$ in the above equation. Since the approximations are applied in opposite directions, one can expect some mutual compensation. Earlier studies Suzuki ; Takano ; Tomczak ; TomRich1 ; TomRich2 have been successful in obtaining finite-temperature behavior on a variety of quantum systems.

The transformation above is summarized by

[TABLE]

where $i,j,k$ are three successive sites. The operator $-\beta^{\prime}\mathcal{H}^{\prime}(i,k)$ acts on two-site states, while the operator $-\beta\mathcal{H}(i,j)-\beta\mathcal{H}(j,k)$ acts on three-site states, so that we can rewrite Eq.(4) in the matrix form,

[TABLE]

where state variables $u,v,w,\bar{u},\bar{v}$ can take spin-up or spin-down values at each site. The unrenormalized $8\times 8$ matrix on the right-hand side is contracted into the renormalized $4\times 4$ matrix on the left-hand side of Eq.(5). We use two-site basis states vectors $\{|\phi_{p}\rangle\}$ and three-site basis states vectors $\{|\psi_{q}\rangle\}$ to diagonalize the matrices in Eq.(5). The states $\{|\phi_{p}\rangle\}$, given in Table I, are eigenstates of parity, total spin magnitude, and total spin z-component. These $\{|\phi_{p}\rangle\}$ diagonalize the renormalized matrix, with eigenvalues

[TABLE]

The states $\{|\psi_{q}\rangle\}$, given in Table II, are eigenstates of parity and total spin z-component. The $\{|\psi_{p}\rangle\}$ diagonalize the unrenormalized matrix, with eigenvalues

[TABLE]

With these eigenstates, Eq.(5) is rewritten as

[TABLE]

Thus, there are three independent $\gamma_{p}$ that determine the renormalized Hamiltonian and, therefore, three renormalized interactions in the Hamiltonian closed under renormalization-group transformation, Eq.(1). These $\gamma_{p}$ are

[TABLE]

which yield the recursion relations

[TABLE]

As expected, $J_{\!xy}^{\prime}$ and $J_{\!z}^{\prime}$ are independent of the additive constant $G$ and the derivative $\partial_{G}G^{\prime}=b^{d}=2$, where $b=2$ is the rescaling factor and $d=1$ is the dimensionality of the lattice.

For $J_{\!xy}=J_{\!z}$, the recursion relations reduce to the spin-$\frac{1}{2}$ isotropic Heisenberg (XXX) model recursion relations, while for $J_{\!xy}=0$ they reduce to spin-$\frac{1}{2}$ Ising model recursion relations. The $J_{\!z}=0$ subspace (XY model) is not (and need not be) closed under these recursion relations Suzuki ; Takano : The renormalization-group transformation induces a positive $J_{\!z}$ value, but the spin-space easy-plane aspect is maintained.

In addition, there exists a mirror symmetry along the $J_{\!z}$-axis, so that $J_{\!xy}^{\prime}\left(-J_{\!xy},J_{\!z}\right)=J_{\!xy}^{\prime}\left(J_{\!xy},J_{\!z}\right)$ and $J_{\!z}^{\prime}\left(-J_{\!xy},J_{\!z}\right)=J_{\!z}^{\prime}\left(J_{\!xy},J_{\!z}\right)$. The thermodynamics of the system remains unchanged under flipping the interactions of the $x$ and $y$ spin components, since the renormalization-group trajectories do not change. In fact, this is part of a more general symmetry of the XYZ model, where flipping the signs of any two interactions leaves the spectrum unchanged Yang . Therefore, with no loss of generality, we take $J_{\!xy}>0$. Independent of the sign of $J_{\!xy}$, $J_{\!z}>0$ gives the ferromagnetic model and $J_{\!z}<0$ gives the antiferromagnetic model.

II.3 Calculation of Densities and Response Functions by the Recursion-Matrix Method

Just as the interaction constants of two consecutive points along the renormalization-group trajectory are related by the recursion relations, the densities are connected by a recursion matrix $\hat{T}$, which is composed of derivatives of the recursion relations. For our Hamiltonian, the recursion matrix and density vector $\vec{M}$ are

[TABLE]

[TABLE]

These are densities $M_{\alpha}$ associated with each interaction $K_{\alpha}$,

[TABLE]

where $N_{\alpha}$ is the number of $\alpha$-type interactions and $Z$ is the partition function for the system, which can be expressed both via the unrenormalized interaction constants as $Z(\vec{K})$ or via the renormalized interaction constants as $Z(\vec{K^{\prime}})$. By using these two equivalent forms, one can formulate the density recursion relation McKay

[TABLE]

Since the interaction constants, under renormalization-group transformation, stay the same at fixed points such as critical fixed points or sinks, the above Eq.(13) takes the form of a solvable eigenvalue equation,

[TABLE]

at fixed points, where $\vec{M}=\vec{M^{\prime}}=\vec{M}^{*}$. The fixed point densities are the components of the left eigenvector of the recursion matrix with left eigenvalue $b^{d}$ McKay . At ordinary points, Eq.(13) is iterated until a sink point is reached under successive renormalization-group transformations. In algebraic form, this means

[TABLE]

where the upper indices indicate the number of iteration (transformation), with $\vec{M}^{(n)}\simeq\vec{M}^{*}$.

This method is applied on our model Hamiltonian. The sink of the system is at infinite temperature $J_{\!xy}^{*}=J_{\!z}^{*}=0$ for all initial conditions $(J_{\!xy},J_{\!z})$.

Response functions are calculated by differentiation of densities. For example, the internal energy is $U=-2\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle-R\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$, employing $T=1/J_{\!xy}$, and $U=-2\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle/R-\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$, employing $T=1/|J_{\!z}|$. The specific heat $C=\partial_{T}U$ follows from the chain rule,

[TABLE]

III Correlations Scanned with Respect to Anisotropy

The ground-state and excitation properties of the XXZ model offer a variety of behaviors Haldane1980 ; Haldane1982 ; Dmitriev ; Takahashi : The antiferromagnetic model with $R<-1$ is Isinglike and the ground state has Néel long-range order along the $z$ spin component with a gap in the excitation spectrum. For $-1\leq R\leq 1$, the system is a ”spin liquid”, with a gapless spectrum and power-law decay of correlations at zero temperature. The ferromagnetic model with $R>1$ is also Isinglike, the ground state is ferromagnetic along the $z$ spin component, with an excitation gap.

Our calculated $\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$ and $\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle\equiv\left\langle s_{i}^{x}s_{j}^{x}\right\rangle=\left\langle s_{i}^{y}s_{j}^{y}\right\rangle$ nearest-neighbor spin-spin correlations for the whole range of the anisotropy coefficient $R$ are shown in Fig.1, for various temperatures. The $xy$ correlation is always non-negative. Recall that we use $J_{\!xy}>0$ with no loss of generality. In the Isinglike antiferromagnetic ($R<-1$) region, the $z$ correlation is expectedly antiferromagnetic. As the $\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$ correlation saturates for large $|R|$, the transverse $\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle$ correlation is somewhat depleted. In the Isinglike ferromagnetic ($R>1$) region, the $\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$ correlation is ferromagnetic, saturates quickly as the $\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle$ correlation quickly goes to zero. In the spin-liquid ($|R|<1$) region, the $\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$ correlation monotonically passes through zero in the feromagnetic side, while the $\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle$ correlation is maximal. The remarkable quantum behavior of $\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$ around $R=0$ is discussed in Sec.V below. It is seen in the figure that these changeovers are increasingly sharp as temperature is decreased and, at zero temperature, become discontinuous at $R=1$. As seen in Fig.1(b), at zero temperature, our calculated $\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$ and $\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle$ correlations show very good agreement with the known exact points Kato ; Kitanine ; Lieb ; Sato . Also, our results for $R>1$ fully overlap the exact results of $\left\langle s_{i}^{z}s_{j}^{z}\right\rangle=0.25$ and $\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle=0$ Takahashi . We also note that zero-temperature is the limit in which our approximation is at its worst.

IV Antiferromagnetic XXZ Chain

For the antiferromagnetic XXZ chain, our calculated $\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$ and $\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle$ nearest-neighbor spin-spin correlations as a function of temperature are shown in Fig.2 for various anisotropy coefficients $R$. We find that when $J_{\!xy}$ is the dominant interaction (spin liquid), the correlations are weakly dependent on anisotropy $R$. When $J_{\!z}$ is the dominant interaction (Isinglike), the correlations are weakly dependent on anisotropy $R$ only at the higher temperatures. Our results are compared with multiple-integral results Bortz in Fig.3.

In every one of the panels of Fig.2, the correlation curves cross each other, revealing a remarkable quantum phenomenon. In a classical system, the correlation between a given spin component (e.g., $\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle$) is expected to decrease when the coupling of another spin component (e.g., $|J_{z}|$) is increased. It is found from the antiferromagnetic XXZ chain in Fig.2 that the opposite may occur in a quantum system: In this figure, an increase in $J_{xy}$ causes an increase in $|\left\langle s_{i}^{z}s_{j}^{z}\right\rangle|$ for $1/|J_{z}|>0.9$ and 0.4 in the spin-liquid and Isinglike regions respectively. Conversely, an increase in $|J_{z}|$ causes an increase in $\left\langle s_{i}^{xy}s_{j}^{xy}\right\rangle$ for $1/J_{xy}>0.4$ and 2.1 in the spin-liquid and Isinglike regions respectively. This quantum effect can be called cross-component spin correlation.

The antiferromagnetic specific heats calculated with Eq.(16) are shown in Fig.4 for various anisotropy coefficients and compared, in Figs.5, 6, with finite-lattice expansion BF ; Narayanan , quantum decimation Xi-Yao , transfer matrix Klumper , high-temperature series expansion Rojas results and, for the $R=0$ case, namely the XY model, with the exact result Katsura $C=(1/4\pi T)\int^{\pi}_{0}\left(\cos\omega/\cosh\left(\frac{\cos\omega}{2T}\right)\right)^{2}d\omega$. The $C(T)$ peak temperature is highest for the isotropic case (Heisenberg) and decreases with anisotropy increasing in either direction (towards Ising or XY). The peak value of $C(T)$ is only weakly dependent on anisotropy, especially for the Isinglike systems. A strong contrast to this behavior will be seen, as another quantum mechanical phenomenon, in the ferromagnetic XXZ chain.

The linearity, at low temperatures, of the spin liquid ($|R|\leq 1$) specific heat with respect to temperature is expected on the basis of spin-wave calculations for the antiferromagnetic XXZ model Kubo ; Kranendonk . This linear form of $C(T)$ reflects the linear energy-momentum dispersion of the low-lying excitations, the magnons. The low-temperature magnon dispersion relation is $\hbar\omega=ck^{n}$, where $c$ is the spin-wave velocity and $n=1$ for the antiferromagnetic XXZ model in $d=1$ Takahashi . The internal energy, given by $U=(1/2\pi)\int_{0}^{\infty}dk\hbar\omega(k)/(e^{\beta\hbar\omega(k)}-1)$, is dominated by the magnons at low temperatures, yielding $U\sim T^{2}$ and $C\sim T$ for $n=1$ in the dispersion relation. From this relation, our calculated spin-wave velocity $c$ as a function of anisotropy $R$ is given in Fig.7 and compares well with the also shown exact result desCloizeaux . A simultaneous fit to the dispersion relation exponent $n$, expected to be 1, yields $1.00\pm 0.02$. However, for the Isinglike $-R>1$, the unexpected linearity instead of an exponential form caused by a gap in the excitation spectrum, points to the approximate nature of our renormalization-group calculation. The correct exponential form is obtained in the large $-R$ limit, where the renormalization-group calculation becomes exact.

Rojas et al. Rojas have obtained the high-temperature expansion of the free energy of the XXZ chain to order $\beta^{3}$, where $\beta$ is the inverse temperature. The specific heat from this expansion is

[TABLE]

This high-temperature specific heat result is also compared with our results, in Fig.6, and very good agreement is seen. In fact, when in the high-temperature region of $0<\beta<0.1$, we fit our numerical results for $C(\beta)$ to the fourth degree polynomial $C=\Sigma_{i=0}^{4}A_{i}\beta^{i}$, and we do find (1) the vanishing $A_{0}<10^{-5}$ and $A_{1}<10^{-7}$ for all $R$ and (2) the comparison in Fig.8 between our results for $A_{2}$ and $A_{3}$ and those of Eq.(17) from Ref.Rojas , thus obtaining excellent agreement for all regions of the model.

V Ferromagnetic XXZ Chain

For the ferromagnetic (i.e., $R>0$) systems in Fig.1, the $\left\langle s_{i}^{z}s_{j}^{z}\right\rangle$ expectation value becomes rapidly negative at lower temperatures for $R<1$, even though for $R\geq 0$ all couplings in the Hamiltonian are ferromagnetic. This is actually a real physical effect, not a numerical anomaly. In fact, we know the spin-spin correlations for the ground state of the one-dimensional XY model (the $R=0$ case of our Hamiltonian), and we can compare our low-temperature results with these exact values. The ground-state properties of the spin-$\frac{1}{2}$ XY model are studied by making a Jordan-Wigner transformation, yielding a theory of non-interacting spinless fermions. Analysis of this theory yields the exact zero-temperature nearest-neighbor spin-spin correlations Lieb shown in Table III. Our renormalization-group results in the zero-temperature limit, also shown in this table, compare quite well with the exact results, as with the other exact points in Fig.1(b), although in the worst region for our approximation. Finally, by continuity, it is reasonable that for a range of $R$ positive but less than one, the $z$ component correlation function is as we find, intriguingly but correctly negative at low temperatures. Thus, the interaction $s_{i}^{x}s_{j}^{x}+s_{i}^{y}s_{j}^{y}$ (irrespective of its sign, due to the symmetry mentioned at the end of Sec.IIB) induces an antiferromagnetic correlation in the $s_{i}^{z}$ component, competing with the $s_{i}^{z}s_{j}^{z}$ interaction when the latter is ferromagnetic.

For finite temperatures, our calculated nearest-neighbor spin-spin correlations are shown in Figs.9, 10, for different values of $R$. These results are compared with Green’s function calculations Zhang in Fig.11. As expected from the discussion at the beginning of this section, in the spin-liquid region, the correlation $\langle s_{i}^{z}s_{j}^{z}\rangle$ is negative at low temperatures. Thus, a competition occurs in the correlation $\langle s_{i}^{z}s_{j}^{z}\rangle$ between the XY-induced antiferromagnetism and the ferromagnetism due to the direct coupling between the $s^{z}$ spin components. In fact, the reinforcement of antiferromagnetic correlations of $\langle s_{i}^{z}s_{j}^{z}\rangle$ by increasing $J_{xy}$ (and also its converse) was seen in the antiferromagnetic XXZ chain discussed in the previous section. Thus, we see that whereas this cross-component effect is dominant at low temperatures in the ferromagnetic XXZ chain, it is seen at higher temperatures in the antiferromagnetic XXZ chain and, in-between, throughout the temperature range in the XY chain.

In the ferromagnetic XXZ chain, as a consequence of the competition mentioned above, a sign reversal in $\langle s_{i}^{z}s_{j}^{z}\rangle$ occurs from negative to positive correlation, at temperatures $J_{xy}^{-1}=T_{0}(R)$.Schindelin At this temperature, by cancelation of the competing effects, the nearest-neighbor correlation $\langle s_{i}^{z}s_{j}^{z}\rangle$ is zero. Our calculated $T_{0}(R)$ curve is shown in Fig.10, and has very good agreement with the exact result $T_{0}=({\sqrt{3}\sin\gamma}/4\gamma)\tan[\pi(\pi-\gamma)/2\gamma]$ where $\gamma\equiv\cos^{-1}(-R)$ Fabricius .

The calculated ferromagnetic specific heats are shown in Fig.12 for various anisotropy coefficients and compared, in Figs.13, 14, with finite-lattice expansion BF , quantum decimation Xi-Yao , decoupled Green’s functions Zhang , transfer matrix Fabricius ; Klumper , high-temperature series expansion Rojas results and, for the $R=0$ case, namely the XY model, with the exact result Katsura $C=(1/4\pi T)\int^{\pi}_{0}\left(\cos\omega/\cosh\left(\frac{\cos\omega}{2T}\right)\right)^{2}d\omega$. In sharp contrast to the antiferromagnetic case in Sec.IV, the peak $C(T)$ temperature is highest for the most anisotropic cases (XY or Ising) and decreases with anisotropy decreasing from either direction (towards Heisenberg). In the same contrast, the peak value of $C(T)$ is dependent on anisotropy, decreasing, eventually to a flat curve, as anisotropy is decreased. This contrast between the ferromagnetic and antiferromagnetic systems is a purely quantum phenomenon. Specifically, the marked contrast between the specific heats of the isotropic antiferromagnetic and ferromagnetic systems, seen in the full curves of Figs.4 and 12 respectively, translates into the different critical temperatures of the respective three-dimensional systems.Rushbrooke ; Oitmaa ; Falicov ; Hinczewski Classical ferromagnetic and antiferromagnetic systems are, on the other hand, identically mapped onto each other.

The low-temperature specifics heats are discussed in detail and compared to other results in Sec.VI.

VI Low-Temperature Specific Heats

Properties of the low-temperature specific heat of the ferromagnetic XXZ chain have been derived from the thermodynamic Bethe-ansatz equations Takahashi . For anisotropy coefficient $|R|\leq 1$, the model is gapless Haldane1980 ; Haldane1982 and, except at $R=1$, the specific heat is linear in $T=J_{xy}^{-1}$ in the zero-temperature limit, $C/T=2\gamma/(3\sin\gamma)$ where again $\gamma\equiv\cos^{-1}(-R)$. Note that this result contradicts the spin-wave theory prediction of $C\sim T^{1/2}$ for the ferromagnetic chain ($n=2$ for the ferromagnetic magnon dispersion relation of the kind given above in Sec.IV). The spin-wave result is valid only for $R=1$, the isotropic Heisenberg case. From the expression given above, we see that $C/T$ diverges as $R\rightarrow 1^{-}$, and at exactly $R=1$ it has been shown that $C\sim T^{1/2}$ Takahashi .

In the Isinglike region $R>1$, the system exhibits a gap in its excitation spectrum and the specific heat behaves as $C\sim T^{-a}\>\exp(-\Delta/T)$, with $\Delta$ being the excitation spectrum gap Haldane1980 ; Haldane1982 ; Takahashi . There exist two gaps for the energy, called the spinon gap and the spin-wave gap, given by $\Delta_{spinon}=\frac{1}{2}\sqrt{1-R^{-2}}$ and $\Delta_{spinwave}=1-R^{-1}$. These are the minimal energies of elementary excitations Takahashi ; Johnson . A crossover between them occurs at $R=\frac{5}{3}$: below this value, the spinon gap is lower, while above this value the spin-wave gap is lower. We have double-fitted our calculated specific heats with respect to the gap $\Delta$ and the leading exponent $a$, for the entire range of anisotropy $R$ between $0<R^{-1}<1$ (Fig.15). Our calculated gap $\Delta$ behaves linearly in $R^{-1}$ for $R^{-1}$ close to 1, and crosses over to 1/2 at $R^{-1}=0$, as expected. We also obtain the exponent $a=1.99\pm 0.02$ in the Ising limit $R^{-1}\leq 0.2$ and $a=1.52\pm 0.10$ in the Heisenberg limit $R^{-1}\geq 0.9$. These exponent values are respectively expected to be 2 and 1.5 Johnsonb ; Johnson .

We now turn to the discussion of our specific heat results for the entire ferromagnetic and antiferromagnetic ranges. Our calculated $C/T$ curves are plotted as a function of anisotropy and temperature in Figs.16 and 17 respectively. We discuss each region of the anisotropy $R$ separately:

(i) $R>1$ : The specific heat coefficient $C/T$ vanishes in the $T\rightarrow 0$ limit and has the expected exponential form as discussed above in this section. The spin-wave to spinon excitation gap crossover is obtained.

(ii) $R\approx 1$ : The double-peak structure of $C/T$ in Fig.16 is centered at $R=1$. As temperature goes to zero, the peaks narrow and diverge.

(iii) $-1\leq R<1$ : The specific heat coefficient is $C/T=2\gamma/(3\sin\gamma)$ in this region Haldane1980 ; Takahashi , and our calculated specific heat is indeed linear at low temperatures. The $C/T$ curves for $R=-1,-0.5,0.5$ in Fig.17 all extrapolate to nonzero limits at $T=0$. The spin-wave dispersion relation exponent and velocity, for the antiferromagnetic system, is correctly obtained for the isotropic case and for all anisotropies, as seen in Fig.7. Fig.18 directly compares $C/T=2\gamma/(3\sin\gamma)$ with our results: The curves have the same basic form, gradually rising from $R=-1$, with a sharp divergence as $R$ nears $1$. At $R=1^{+}$, we expect $C/T=0$. Our $T=10^{-10}$ curve diverges at $R=1$ and indeed returns to zero at $R=1.0000001$.

(iv) $R<-1$ : We expect a vanishing $C/T$, which we do find as seen in Fig.16 and in the insets of Fig.17. The exponential behavior of the specific heat is clearly seen in the Ising limit.

VII Conclusion

A detailed global renormalization-group solution of the XXZ Heisenberg chain, for all temperatures and anisotropies, for both ferromagnetic and antiferromagnetic couplings, has been obtained. In the spin-liquid region, the linear low-temperature specific heat and, for the antiferromagnetic chain, the spin-wave dispersion relation exponent $n$ and velocity $c$ have been obtained. In the Isinglike region, the spin-wave to spinon crossover of the excitation spectrum gap of the ferromagnetic chain has been obtained from the exponential specific heat, as well as the correct leading algebraic behaviors in the Heisenberg and Ising limits. Purely quantum mechanical effects have been seen: We find that the $xy$ correlations and the antiferromagnetic $z$ correlations mutually reinforce each other, for different ranges of temperatures and anisotropies, in ferromagnetic, antiferromagnetic, and XY systems. The behaviors, with respect to anisotropy, of the specific heat peak values and locations are opposite in the ferromagnetic and antiferromagnetic systems. The sharp contrast found in the specific heats of the isotropic ferromagnetic and antiferromagnetic systems is a harbinger of the different critical temperatures in the respective three-dimensional systems. When compared with existing calculations in the various regions of the global model, good quantitative agreement is seen. Even at zero temperature, where our approximation is at its worst, good quantitative agreement is seen with exact data points for the correlation functions (Fig.1(b)), which we extend to all values of the anisotropy. Finally, the relative ease with which the Suzuki-Takano decimation procedure is globally and quantitatively implemented should be noted.

Acknowledgements.

This research was supported by the Scientific and Technological Research Council (TÜBİTAK) and by the Academy of Sciences of Turkey. One of us (O.S.S.) gratefully acknowledges a scholarship from the Turkish Scientific and Technological Research Council - Scientist Training Group (TÜBİTAK-BAYG).