The ground state-vector of the XY Heisenberg chain and the Gauss decomposition

TL;DR

This paper derives an exact group-theoretical expression for the ground state-vector of the XY Heisenberg spin chain in fermion representation, facilitating advanced analysis of its correlation functions.

Contribution

It provides a novel exact formula for the ground state-vector using group theory, enabling deeper study of correlation functions in the XY model.

Findings

Exact expression for the ground state-vector derived

Facilitates combinatorial analysis of correlation functions

Enhances understanding of the XY Heisenberg chain

Abstract

The XY Heisenberg spin 1/2 chain is considered in the fermion representation. The construction of the ground state-vector is based on the group-theoretical approach. The exact expression for the ground state-vector will allow to study the combinatorics of the correlation functions of the model.

Equations

$$H=H_{\mathrm{xx}}+\gamma H_{\mathrm{anis}},H_{\mathrm{xx}}\equiv -\sum _{n,m=1}^M{\Delta }_{nm}^{(+)}{\sigma }_n^{+}{\sigma }_m^{-},$$

$$H_{\mathrm{anis}}\equiv -\frac{1}{2}\sum _{n,m=1}^M{\Delta }_{nm}^{(+)}({\sigma }_n^{+}{\sigma }_m^{+}+{\sigma }_n^{-}{\sigma }_m^{-}),$$

$$[{\sigma }_k^{+},{\sigma }_l^{-}]={\delta }_{kl}{\sigma }_l^z,[{\sigma }_k^z,{\sigma }_l^{\pm }]=\pm 2{\delta }_{kl}{\sigma }_l^{\pm }.$$

$${\Delta }_{nm}^{(s)}\equiv \frac{1}{2}({\delta }_{|n-m|,1}+s{\delta }_{|n-m|,M-1}),$$

$$\{c_k,c_n\}=\{c_k^{†},c_n^{†}\}=0,\{c_k,c_n^{†}\}={\delta }_{kn},$$

$$c_{k}=\exp\Big{(}i\pi\sum\limits^{k-1}_{n=1}\sigma^{-}_{n}\sigma^{+}_{n}\Big{)}\,\sigma^{+}_{k},\qquad c^{\dagger}_{k}=\sigma^{-}_{k}\,\exp\Big{(}\!-i\pi\sum\limits^{k-1}_{n=1}\sigma^{-}_{n}\sigma^{+}_{n}\Big{)}\,.$$

$${\sigma }_n^{-}=c_n^{†}\prod _{j=1}^{n-1}(1-2c_j^{†}{c}_j),{\sigma }_n^{+}=\prod _{j=1}^{n-1}(1-2c_j^{†}{c}_j)c_n.$$

$$c_{M+1}=(-1{)}^{\mathcal{N}}{c}_1,c_{M+1}^{†}=c_1^{†}(-1{)}^{\mathcal{N}},$$

$$H=$$

$$H^{\pm }=$$

$$c_{M+1}=-s{c}_1,c_{M+1}^{†}=-s{c}_1^{†}.$$

$$c_k=\frac{e^{-i\pi /4}}{\sqrt{M}}\sum _{q\in {\mathsf{S}}^{\pm }}{e}^{iqk}{c}_q,c_k^{†}=\frac{e^{i\pi /4}}{\sqrt{M}}\sum _{q\in {\mathsf{S}}^{\pm }}{e}^{-iqk}{c}_q^{†},$$

$${\mathsf{S}}^{+}=\{q:q=-\pi +\pi (2l-1)/M,l\in \mathcal{M}\},$$

$${\mathsf{S}}^{-}=\{q:q=-\pi +2\pi l/M,l\in \mathcal{M}\}.$$

$$H^{\pm }=\frac{1}{2}\sum _{q\in {\mathsf{S}}^{\pm }}(c_q^{†},c_{-q}){\mathcal{H}}_q\left(\begin{array}{c}{c}_q\\ c_{-q}^{†}\end{array}\right),$$

$${\mathcal{H}}_q={ϵ}_q{\sigma }^z+{\Gamma }_q({\sigma }^{+}+{\sigma }^{-}),$$

$${\mathcal{J}}_q^{-}=c_{-q}{c}_q,{\mathcal{J}}_q^{+}=c_q^{†}{c}_{-q}^{†},$$

$${\mathcal{J}}_q^0=\frac{1}{2}(c_q^{†}{c}_q+c_{-q}^{†}{c}_{-q}-1).$$

$$[{\mathcal{J}}_q^{+},{\mathcal{J}}_p^{-}]=2{\mathcal{J}}_q^0{\delta }_{pq},[{\mathcal{J}}_q^0,{\mathcal{J}}_p^{\pm }]=\pm {\mathcal{J}}_q^{\pm }{\delta }_{pq}.$$

$$H^{\pm }=\sum _{q\in {\mathsf{S}}^{\pm }}{ϵ}_q{\mathcal{J}}_q^0+\frac{{\Gamma }_q}{2}({\mathcal{J}}_q^{+}+{\mathcal{J}}_q^{-}).$$

$$\left(\begin{array}{c}{c}_q\\ c_{-q}^{†}\end{array}\right)=g_{\theta }^{†}\left(\begin{array}{c}{A}_q\\ A_{-q}^{†}\end{array}\right),$$

$$g_{\theta }^{†}=e^{-{\theta }_q({\sigma }^{-}-{\sigma }^{+})}=e^{i{\theta }_q{\sigma }^y}=\cos {\theta }_q{\sigma }^0+\sin {\theta }_q({\sigma }^{+}-{\sigma }^{-}).$$

$$g_{\theta }{\mathcal{H}}_q{g}_{\theta }^{†}=E_q{\sigma }^z.$$

$${ϵ}_q\cos 2{\theta }_q-{\Gamma }_q\sin 2{\theta }_q=E_q,$$

$${ϵ}_q\sin 2{\theta }_q+{\Gamma }_q\cos 2{\theta }_q=0,$$

$${\bar{\mathcal{J}}}_q^{-}=A_{-q}{A}_q,{\bar{\mathcal{J}}}_q^{+}=A_q^{†}{A}_{-q}^{†},$$

$${\bar{\mathcal{J}}}_q^0=\frac{1}{2}(A_q^{†}{A}_q+A_{-q}^{†}{A}_{-q}-1).$$

$${\mathcal{J}}_q^{-}+{\mathcal{J}}_q^{+}=({\bar{\mathcal{J}}}_q^{-}+{\bar{\mathcal{J}}}_q^{+})\cos 2{\theta }_q-2{\bar{\mathcal{J}}}_q^0\sin 2{\theta }_q,$$

$$2{\mathcal{J}}_q^0=({\bar{\mathcal{J}}}_q^{-}+{\bar{\mathcal{J}}}_q^{+})\sin 2{\theta }_q+2{\bar{\mathcal{J}}}_q^0\cos 2{\theta }_q.$$

$$H^{\pm }=\sum _{q\in {\mathsf{S}}^{\pm }}{E}_q{\bar{\mathcal{J}}}_q^0=\sum _{q\in {\mathsf{S}}^{\pm }}{E}_q{A}_q^{†}{A}_q+E_{\mathrm{gr}}^{\pm },$$

$$E_{\mathrm{gr}}^{\pm }=\frac{-1}{2}\sum _{q\in {\mathsf{S}}^{\pm }}{E}_q.$$

$$c_k|0⟩=0,⟨0|c_k^{†}=0,k\in \mathcal{M},$$

$$c_q|0⟩=0,⟨0|c_q^{†}=0,q\in {\mathsf{S}}^{\pm }.$$

$$A_q|0⟩⟩=0,⟨⟨0|A_q^{†}=0,q\in {\mathsf{S}}^{\pm }.$$

$${\cal U}_{\theta}\,\equiv\,\exp\Bigl{(}\sum_{q\in{\sf S}^{\pm}}{\theta_{q}}({{\cal J}}^{+}_{q}-{{\cal J}}^{-}_{q})\Bigr{)}=\prod_{q\in{\sf S}^{\pm}}\exp\bigl{(}{{\theta_{q}}({{\cal J}}^{+}_{q}-{{\cal J}}^{-}_{q})}\bigr{)}\,,$$

Full text

Key words: $XY$ Heisenberg spin chain, ground state-vector, Gauss decomposition

1 Introduction

The correlation functions of certain quantum integrable models demonstrate connection with enumerative combinatorics and with the theory of symmetric functions [1, 2, 3, 4, 5]. For instance, random lattice walks and boxed plane partitions, as subjects of enumerative combinatorics [6], are related to the correlation functions of the $XX$ model [7, 8, 9, 10, 11]. Various spin lattice models [12], including the $XY$ Heisenberg chain model, as well as its isotropic limit, the $XX$ model, provide a base for such actively developing subjects in the theory of quantum information [13] as random lattice walks [14] and entanglement entropy [15].

Interest in the study of the correlation functions for the $XY$ spin chain still exists after the pioneer works [16, 17, 18]. The determinantal representation for the the equal-time correlation functions for the model was obtained in the paper [19]. The approach based on the application of the coherent states to the problem of time and temperature dependent correlation functions was developed in [20, 21]. In the present paper we consider a group theoretical approach to the $XY$ model and derive its ground state-vector using the Gauss decomposition [22]. The representation of the obtained ground state wave function would be of importance in the combinatorial interpretation of the correlation functions of the model. Namely, the scalar products of the state-vectors may be described as a linear combination of nests of self-avoiding lattice paths, so-called, watermelons (see Figure 1 and Ref. [2]).

The paper is organised as follows. Section 1 is introductory. The fermion representation of the model is provided in Section 2, and diagonalization of the Hamiltonian by means of the Bogoliubov transformation is performed. The ground state wave function is derived in Section 3. Section 4 concludes the paper.

2 Outline of the model

The $XY$ Heisenberg spin chain is described by the Hamiltonian, [16, 17]:

[TABLE]

where $\gamma$ is anisotropy parameter, and $M=0\pmod{2}$ is the number of sites. The local spin operators $\sigma^{\pm}_{n}=\frac{1}{2}(\sigma^{x}_{n}\pm i\sigma^{y}_{n})$ and $\sigma^{z}_{n}$ depend on the lattice argument $n\in\mathcal{M}\equiv\{1,2,\dots,M\}$ and satisfy the commutation relations:

[TABLE]

The entries of the hopping matrix $\Delta^{(s)}$ in (1), (2) are expressed as follows, [9]:

[TABLE]

where ${\delta}_{n,l}(\equiv{\delta}_{nl})$ is the Kronecker symbol, and $s$ is either $\pm 1$ or zero. The periodic boundary conditions $\sigma^{\alpha}_{n+M}=\sigma^{\alpha}_{n}$, $\forall n\in\mathcal{M}$, are imposed. The Hamiltonian $H_{\rm xx}$ (1) (i.e., $H$ at $\gamma=0$) is that of the periodic $XX$ chain.

Let us pass from the spin operators $\sigma^{\alpha}_{k}$ to the canonical fermion operators $c_{k}$, $c^{\dagger}_{k}$ subjected to the algebra

[TABLE]

where the brackets $\{\,,\,\}$ imply anti-commutation. We use the Jordan-Wigner transformation [23]:

[TABLE]

Inversion of (6) takes the form:

[TABLE]

The periodic boundary conditions for the spin variables are equivalent to the following boundary conditions for the fermion variables:

[TABLE]

where ${\mathcal{N}}\equiv Q(M)=\sum^{M}_{k=1}c^{\dagger}_{k}c_{k}$ is the total number of particles.

The transformations (6), (7) enable to represent $H$ (1) as follows [16, 17]:

[TABLE]

where $P^{\pm}$ are the projectors onto the states with even ($+$)/odd ($-$) number of fermions: $P^{\pm}$ $=$ $\frac{1}{2}(1\pm(-1)^{\mathcal{N}})$. The indices $s=\pm$ point out the correspondence between the operators $H^{s}$ (10) and appropriately specified boundary conditions (8):

[TABLE]

The operator ${\mathcal{N}}$ commutes with $H_{\rm xx}$ (1). The parity operator $(-1)^{{\mathcal{N}}}$ commutes with $H$ and anti-commutes with $c^{\dagger}_{k}$ and $c_{k}$.

The requirements (11) suggest to use the Fourier series:

[TABLE]

where $\sum_{q\in{\sf S}^{\pm}}$ implies summation over quasi-momenta $q\in{\sf S}^{\pm}$ respecting $\cos Mq=\mp 1$:

[TABLE]

Substitution of (12) into (10) yields the $XY$ Hamiltonian in the momentum representation:

[TABLE]

where $\epsilon_{q}\equiv-\cos q$ and $\Gamma_{q}\equiv\gamma\sin q$.

It appropriate to introduce three quadratic operators ${\cal J}^{\pm}_{q}$, ${\cal J}_{q}^{0}$ expressed through the fermion operators $c_{q}$ and $c^{\dagger}_{q}$:

[TABLE]

The operators (15), (16) are related to the algebra $\mathfrak{su}(2)$ since satisfy the commutation relations of the form (compare with (3)):

[TABLE]

The definitions (15) and (16) allow us to express $H^{\pm}$ (13) as follows:

[TABLE]

Let us relate the canonical operators $c^{\dagger}_{q}$, $c_{q}$ to the new fermionic operators $A^{\dagger}_{q}$, $A_{q}$ by means of the unitary matrix $g^{\dagger}_{\theta}\in SU(2,\mathbb{R})$,

[TABLE]

The transformation (19) and its conjugated are used in (13), and it enables to diagonalize the matrix ${\cal H}_{q}$ (14) as follows:

[TABLE]

The relation (21) is equivalent to the following equations:

[TABLE]

where $E_{q}=(\epsilon_{q}^{2}+\Gamma^{2}_{q})^{1/2}$. It follows from (23) that $\theta_{q}$ respects $\tan 2\theta_{q}=-\Gamma_{q}/\epsilon_{q}$.

Let us introduce, analogously to (15), (16), the appropriate operators $\bar{\cal J}^{\pm}_{q}$, $\bar{\cal J}_{q}^{0}$ in terms of the fermion operators $A_{q}$ and $A^{\dagger}_{q}$:

[TABLE]

The following transformation takes place:

[TABLE]

Applying the transformations (26) and (27) allows us to express the Hamiltonians $H^{\pm}$ (18) as follows:

[TABLE]

provided that the relations (22) and (23) hold, and $E^{\pm}_{\rm{gr}}$ is expressed as

[TABLE]

3 The ground state wave function

The canonical operators $c_{k}$, $c^{\dagger}_{k}$, as well as $c_{q}$, $c^{\dagger}_{q}$, characterized by the relations (5) possess the Fock vacuum $|0\rangle$ (and its conjugate $\langle 0|$):

[TABLE]

The vacuum $|0\rangle$ is normalized, $\langle 0|0\rangle=1$, and $|0\rangle$ is the same for both Hamiltonians $H^{\pm}$.

The ground-state vector $|0\rangle\rangle$ of the Hamiltonian (28) have to satisfy the relations:

[TABLE]

We introduce the unitary operator ${\cal U}_{\theta}$,

[TABLE]

where ${\cal J}^{-}_{q}$ and ${\cal J}^{+}_{q}$ are defined by (15) and 17) and formulate the following

**Proposition 1  ** The relations (31) take place provided that the state $|0\rangle\rangle$ is defined as:

[TABLE]

**Proof  ** The commutation relations are valid:

[TABLE]

where the property $\theta_{-q}=-\theta_{q}$ is used. Taking into account (34) and (35) we obtain:

[TABLE]

The state $|0\rangle\rangle$ (33) is annihilated by $A_{q}$ (36) since $c_{q}$ annihilates the Fock vacuum $|0\rangle$, Eq. (30), and ${\cal U}^{\dagger}_{\theta/2}\,{\cal U}_{\theta/2}=1$. The introduced ground state-vector (33) is normalized to unity, $\langle\langle 0|0\rangle\rangle=1$.

The alternative derivation of Proposition 1 is based on the equivalence of the relations (36), (37) and the transformation (19). The action of operator $A_{q}$ on the ground state (33) may be written as

[TABLE]

The commutation relations

[TABLE]

ensure that

[TABLE]

$\,\,\Box$

The Gauss decomposition, [22], which may be obtained by means of ‘infinitesimal method’ [24], is valid for the matrix g_{\theta}\equiv\exp(-i\theta\sigma^{y})=\exp\bigl{(}\theta(\sigma^{-}-\sigma^{+})\bigr{)} (20):

[TABLE]

where $e^{\bar{\gamma}}=1/\cos\theta$. With regard to (42) we arrive at the decomposition for the elements of the operator ${\cal U}_{\theta}$ (32):

[TABLE]

where $e^{{\bar{\gamma}}_{q}}=1/\cos\theta_{q}$. Equation (43) suggests to formulate the following

**Proposition 2  ** Provided that the representation (43) holds, the state $|0\rangle\rangle$ (33) acquires the equivalent representation:

[TABLE]

**Proof  ** First of all, the following note is of importance:

[TABLE]

where $\sum_{q^{+}}\equiv\sum_{\{(q\in{\sf S}^{\pm})\cap\,(q\in\mathbb{R}^{+})\}}$, and antisymmetry of ${\theta_{q}}$, ${{\cal J}}^{+}_{q}$, ${{\cal J}}^{-}_{q}$ with respect to the reflection of $q$ is taken into account in (45). The Gauss decomposition (43) is used to pass from (45) to (46).

From definitions of operators ${\cal J}^{-}_{q}$ (15) and ${{\cal J}}^{0}_{q}$ (16) it follows that ${\cal J}^{-}_{q}|0\rangle=0$, while ${{\cal J}}^{0}_{q}|0\rangle=\frac{-1}{2}|0\rangle$. Therefore,

[TABLE]

and

[TABLE]

Thus, we obtain from (46), (47) and (48) that (44) is valid. $\,\,\Box$

The representation (44) of the ground state $|0\rangle\rangle$ coincides with that proposed in [20]:

[TABLE]

The normalizing factor $N^{-1/2}$, where $N=\langle 0|\,\Omega\,\Omega^{+}\,|0\rangle$ was calculated in [20] as the integral over the Grassmann coherent states, is equal to

[TABLE]

and coincides with the coefficient in (44). The statement of Proposition 2 clarifies the origin of this coefficient from the group theoretical viewpoint.

For the sake of completeness we shall give the direct proof that the state $|0\rangle\rangle$ expressed by (44) is annihilated by the operator $A_{q}$. Really, since $A_{q}$ is given by (36), it is enough to show that the state

[TABLE]

is annihilated by $c_{q}$. The Gauss decomposition (46) admits the ‘‘antinormal’’ from looking as follows (see [22]):

[TABLE]

Substituting the conjugated form of (51) into (50) we obtain the state annihilated by $c_{q}$:

[TABLE]

Let us turn to the state $|0\rangle\rangle$ (44) and consider the following representation:

[TABLE]

where $\sum_{\{q^{+}_{l}\}_{1\leq l\leq n}}\equiv\sum_{q^{+}_{1},\,q^{+}_{2},\ldots,\,q^{+}_{n}}$, and the sum over $n$ is finite since ${{\cal J}}^{+}_{q}$ squared is zero. The expression in right-hand side of (52) is similar to that derived in [19] as the ground state wave function of $XY$ chain. Recall that $\tan 2\theta_{q}=-\Gamma_{q}/\epsilon_{q}$ (see (23)) is known, and therefore $\tan\theta_{q}$ is found in the form:

[TABLE]

where $T_{q}$ is defined in (52). The answer (53) fulfils the quadratic equation as the corresponding parameter does in [19]. With regard to (53) it can be argued that Eq. (52) just coincides (up to irrelevant factor) with the ground state wave function found in [19].

4 Conclusion

The expression for the ground ground state-vector (33) of the $XY$ spin chain obtained by the group theoretical approach was written in the form (44) with the help of Gaussian decomposition. The representation (44) brought to the form (52) reveals the connection between state-vectors studied in [19] and [20].

The approach discussed in the present paper will allow to spread the combinatorial interpretation of the correlation functions developed in [2] for the $XX$ model on the $XY$ case.

Acknowledgement

This work was supported by the Russian Science Foundation (grant no. 18-11-00297).