Goldstone modes and the golden spiral in the ferromagnetic spin-1 biquadratic model

TL;DR

This paper explores the rich structure of ferromagnetic ground states in the spin-1 biquadratic model, revealing scale invariance, Goldstone modes, Fibonacci-Lucas degeneracy sequences, and logarithmic entanglement entropy scaling.

Contribution

It uncovers the scale invariance and degeneracy patterns of ferromagnetic ground states, linking them to Fibonacci-Lucas sequences and the golden spiral, with detailed analysis of entanglement entropy.

Findings

Ground state degeneracies follow Fibonacci-Lucas sequences.

Goldstone modes depend on system size parity.

Entanglement entropy scales logarithmically with block size.

Abstract

Ferromagnetic ground states have often been overlooked in comparison to seemingly more interesting antiferromagnetic ground states. However, both the physical and mathematical structure of ferromagnetic ground states are particularly rich. We show that the highly degenerate and highly entangled ground states of the ferromagnetic spin-1 biquadratic model are scale invariant, originating from spontaneous symmetry breaking from ${\rm SU}(3)$ to ${\rm U}(1)\times {\rm U}(1)$ with two type-B Goldstone modes if the system size is even or from ${\rm SU}(2)$ to ${\rm U}(1)$ with one type-B Goldstone mode if the system size is odd, when periodic boundary conditions are adopted. The ground state degeneracies are characterized as Fibonacci-Lucas sequences, under open and periodic boundary conditions, with nonzero residual entropy per site. This implies that the ground state degeneracies for this model are asymptotically the golden spiral. In addition, sequences of atypical (periodic) degenerate ground states generated from highest and generalized highest weight states are constructed to establish that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit. The prefactor is half the number of type-B Goldstone modes, which is identified to be the fractal dimension, if one is restricted to atypical degenerate ground states. We also argue that the same conclusion is valid for typical (non-periodic) degenerate ground states, as long as the block size is sufficiently large.

Equations

$$N_A+2N_B=N_{BG},$$

$$S(n)=\frac{N_B}{2}{\log }_{2}n+S_0.$$

$$\mathcal{H}=\sum _j({\text{S}}_j\cdot {\text{S}}_{j+1}{)}^2.$$

$$\dim ({\Omega }_L^{\mathrm{OBC}})$$

$$\dim ({\Omega }_L^{\mathrm{PBC}})$$

$$|L,M_1,M_2{⟩}_p=\frac{1}{Z_p(L,M_1,M_2)}{F}_1^{M_1}{F}_2^{M_2}|s_1{s}_2...s_p...s_1{s}_2...s_p⟩,$$

$$|L,M_1,M_2{⟩}_2=\frac{1}{Z_2(L,M_1,M_2)}{F}_1^{M_1}{F}_2^{M_2}|1...1⟩,$$

$$Z_2(L,M_1,M_2)=M_1!M_2!\sqrt{C_{L/2}^{M_1}{C}_{L-M_1}^{M_2}}.$$

$$|L,M_1,M_3{⟩}_2=\frac{1}{Z_2(L,M_1,M_3)}{F}_1^{M_1}{F}_3^{M_3}|1...1⟩,$$

$$Z_2(L,M_1,M_3)=M_1!M_3!\sqrt{C_{L/2}^{M_3}{C}_{L/2+M_3}^{M_1}}.$$

$$|L,M_2,M_3{⟩}_4=\frac{1}{Z_4(L,M_2,M_3)}{F}_2^{M_2}{F}_3^{M_3}|1110...1110⟩,$$

$$Z_4(L,M_2,M_3)=M_2!M_3!\sqrt{C_{L/4}^{M_3}{C}_{3L/4-M_3}^{M_2}}.$$

$$|L,M_1,M_2{⟩}_6=$$

$$F_1^{M_1}{F}_2^{M_2}|10-1-\mathrm{101...10}-1-101⟩,$$

$$Z_6(L,M_1,M_2)=M_1!M_2!\sqrt{\sum _{l=0}^{M_1}{C}_{L/6}^l{C}_{L/6}^{M_1-l}{C}_{L/3-l}^{M_2}}.$$

$$|L,M_1,M_2{⟩}_p=\sum _{k_1}\sum _{k_2}\lambda (L,n,k_1,k_2,M_1,M_2)|n,k_1,k_2{⟩}_p|L-n,M_1-k_1,M_2-k_2{⟩}_p,$$

$$\lambda (L,n,k_1,k_2,M_1,M_2)=C_{M_1}^{k_1}{C}_{M_2}^{k_2}\frac{Z_p(n,k_1,k_2)Z_p(L-n,M_1-k_1,M_2-k_2)}{Z_p(L,M_1,M_2)}.$$

$$S_L(n,M_1,M_2)=-\sum _{k_1,k_2}\Lambda (L,n,k_1,k_2,M_1,M_2){\log }_2\Lambda (L,n,k_1,k_2,M_1,M_2),$$

Full text

I introduction

As a basic notion, spontaneous symmetry breaking (SSB) plays a prominent role in atomic and particle physics, condensed matter physics, astrophysics and cosmology andersonbook ; SSBbook . In particular, if a broken symmetry group is continuous, a gapless Goldstone mode (GM) emerges. According to Goldstone goldstone ; goldstone2 ; goldstone3 , the number of GMs is equal to the number of broken symmetry generators $N_{BG}$ for a relativistic system undergoing SSB. However, for a nonrelativistic system it becomes much more complicated to clarify the connection between $N_{BG}$ and the number of GMs, as reflected in early work by Nielsen and Chadha nielsen . Since then, much effort has been made to establish a proper classification of GMs schafer ; miransky ; nicolis ; nicolis2 ; brauner-watanabe ; watanabe ; watanabe2 ; NG ; NG2 ; NG3 . A relevant point is the early debate between Anderson anderson and Peierls peierls1 ; peierls2 regarding whether or not a ferromagnetic state should be considered as a result of SSB, now understood to be a paradigmatic example of SSB. An insightful observation made by Nambu nambu led to the classification of type-A and type-B GMs watanabe ; watanabe2 ; NG ; NG2 ; NG3 by introducing the Watanabe-Brauner matrix brauner-watanabe . As a result, when the symmetry group $G$ is spontaneously broken into $H$, the counting rule for the GMs is established as

[TABLE]

where $N_{A}$ and $N_{B}$ are the numbers of type-A and type-B GMs, and $N_{BG}$ is equal to the dimension of the coset space $G/H$.

A recent further development concerns a connection between the entanglement entropy and the counting rule (1) for both type-A Metlitski ; Rademaker and type-B shiqianqian ; dyw GMs. In particular, the entanglement entropy $S(n)$ for the highly degenerate ground states scales logarithmically, in the thermodynamic limit, with the block size $n$ according to

[TABLE]

The prefactor is precisely half the number of type-B GMs $N_{B}$, thus leading to the identification of the number of type-B GMs with the fractal dimension $d_{f}$ introduced doyon from a field theoretic perspective, namely $d_{f}=N_{B}$. As a consequence, previous findings about the entanglement entropy for the $\rm{SU}(2)$ ferromagnetic states are reproduced popkov1 ; popkov2 . This indicates that the highly degenerate ground states arising from SSB with type-B GMs are scale, but not conformally, invariant, reflecting the fact that there is an abstract fractal that lives in a Hilbert space. In a sense, such an interpretation for a ferromagnetic state as a scale invariant state challenges the folklore that scale invariance implies conformal invariance, as first put forward by Polyakov polyakov .

Insights towards understanding such unexpected connections can be gained from exactly solvable quantum many-body systems baxterbook ; giamarchi ; sutherlandb ; mccoy ; sachdev . An example is the spin-1 bilinear-biquadratic model chubukov ; fath1 ; fath2 ; fath3 ; kawashima ; ivanov ; rizzi ; schmid ; porras ; romero ; rakov ; sierra1 ; sierra2 ; ronny ; lundgren ; dyw , widely studied due to its prominent position in conceptual developments in condensed matter physics. A possible realization of the spin-1 bilinear-biquadratic model has been proposed via ultracold atoms in optical lattices rodriguez ; chiara ; greiner1 ; jodens ; Schneider . The model includes the Affleck-Kennedy-Lieb-Tasaki (AKLT) point aklt1 ; aklt2 , which is a key to unlocking the mysteries behind the Haldane phase in the vicinity of the spin-1 antiferromagnetic Heisenberg model – a prototypical example for symmetry-protected topological phases spt1 ; spt2 ; spt3 ; spt4 . The general model also includes other exactly solvable points: the $\rm{SU(3)}$ ferromagnetic and antiferromagnetic models uimin ; lai ; sutherland2 , the spin-1 Takhtajan-Babujian model tb1 ; tb2 and the ferromagnetic and antiferromagnetic spin-1 biquadratic models barber ; klumper . Among them the latter models are also of interest because they constitute a representation of the Temperley-Lieb algebra tla ; martin ; baxterbook , which in turn is relevant to the Jones polynomial in knot theory jone . Each of the spin-1 Uimin-Lai-Sutherland, Takhtajan-Babujian and biquadratic models fit within the powerful framework of Yang-Baxter integrability in mathematical physics baxterbook ; sutherlandb ; mccoy . Moreover, a spin-1 biquadratic model is mapped to a nine-state Potts model barber , but the underlying physics is quite different. They share the same energy spectrum, but not level degeneracies, due to different symmetries.

For the $\rm{SU(3)}$ ferromagnetic spin-1 model uimin ; lai ; sutherland2 , there is a sequence of highly degenerate and highly entangled ground states popkov1 ; popkov2 , as a result of SSB from $\rm{SU(3)}$ to $\rm{SU(2)}\times\rm{U(1)}$, with two type-B GMs shiqianqian , if a ground state, over which the expectation values of the local order parameters are taken, is restricted to the highest weight state. Thus the SSB pattern from $\rm{SU(3)}$ to $\rm{SU(2)}\times\rm{U(1)}$ occurs in the $\rm{SU(3)}$ ferromagnetic spin-$1$ model shiqianqian , resulting in four broken generators, accompanied with two type-B GMs. Hence the counting rule (1) is satisfied, with $N_{A}=0$, $N_{B}=2$ and $N_{BG}=4$.

As noted above, both the uniform $\rm{SU(3)}$ ferromagnetic spin-1 model uimin ; lai ; sutherland2 and the staggered $\rm{SU(3)}$ ferromagnetic spin-1 biquadratic model are two exactly solvable points of the more general spin-1 bilinear-biquadratic model. Given they share $\rm{SU(2)}$ ferromagnetic states as degenerate ground states, one may anticipate that SSB with type-B GMs also emerges in the ferromagnetic spin-1 biquadratic model. As we shall see, the SSB pattern is from $\rm{SU(3)}$ to $\rm{U(1)}\times\rm{U(1)}$. In fact, the degenerate ground states admit an exact singular value decomposition, leading to the unveiling of the underlying self-similarities and implying the existence of a fractal characterized in terms of the fractal dimension $d_{f}$ doyon . A scaling analysis of the entanglement entropy (2) makes it possible to identify the fractal dimension $d_{f}$ with the number of the type-B GMs $N_{B}$. In this case we find that $d_{f}=N_{B}$, with $N_{B}=2$.

For the $\rm{SU(3)}$ ferromagnetic spin-$1$ biquadratic model, six of the eight generators are spontaneously broken, implying that the two type-B GMs originate from four broken generators, with the two other generators being redundant, as required to keep consistency with the counting rule. That is, there is an apparent difference between the two SSB patterns, from $\rm{SU(3)}$ to $\rm{SU(2)}\times\rm{U(1)}$ and from $\rm{SU(3)}$ to $\rm{U(1)}\times\rm{U(1)}$. The former is an illustrative example for the counting rule watanabe , with four broken generators yielding two type-B GMs. In contrast, for the latter, six generators are spontaneously broken, thus apparently violating the counting rule. This justifies the necessity to introduce the redundancy principle for type-B GMs, similar to SSB with type-A GMs typeared .

The finite-size ground states for the ferromagnetic spin-1 biquadratic model are highly degenerate, with the degeneracies being exponential with the system size. As a consequence, the residual entropy is non-zero spins . We find that the ground state degeneracies constitute the two Fibonacci-Lucas sequences, depending on either open boundary conditions (OBC) or periodic boundary conditions (PBC). As a result the ground state degeneracies are asymptotically the golden spiral – an extremely well known self-similar geometric object.

We turn now to the details of our calculations.

II The ferromagnetic spin-1 biquadratic model

The ferromagnetic spin-1 biquadratic model is described by the Hamiltonian

[TABLE]

Here $\textbf{S}_{j}=(S_{x,j},S_{y,j},S_{z,j})$, with $S_{x,j}$, $S_{y,j}$ and $S_{z,j}$ being the spin-1 operators at lattice site $j$. The sum over $j$ is taken from 1 to $L-1$ for OBC, and from 1 to $L$ for PBC. Unless otherwise stated, we assume that the system size $L$ is even. Generically, the model (3) possesses the staggered symmetry group ${\rm SU(3)}$, with its generators being the eight traceless matrices $J_{\eta}=\sum_{j}J_{\eta,j}$, for $\eta=1,\ldots,8$, where $J_{1,j}=S_{x,j}/2$, $J_{2,j}=S_{y,j}/2$, $J_{3,j}=S_{z,j}/2$, $J_{4,j}=(-1)^{j}(1-3/2{S_{z,j}}^{2})$, $J_{5,j}=(-1)^{j}({S_{x,j}}^{2}-{S_{y,j}}^{2})/2$, $J_{6,j}=(-1)^{j}(S_{x,j}S_{y,j}+S_{y,j}S_{x,j})/2$, $J_{7,j}=(-1)^{j}(S_{y,j}S_{z,j}+S_{z,j}S_{y,j})/2$, and $J_{8,j}=(-1)^{j}(S_{z,j}S_{x,j}+S_{x,j}S_{z,j})/2$ cfJxyz .

The exact solvability barber ; klumper of the spin-1 biquadratic model originates from the Hamiltonian (3) constituting (up to an additive constant) a representation of the Temperley-Lieb algebra tla ; baxterbook ; martin , and follows from a solution to the Yang-Baxter equation baxterbook ; sutherlandb ; mccoy . A peculiar feature of the model is that the ground states are highly degenerate, exponential with the system size $L$, thus leading to non-zero residual entropy spins .

III SSB with type-B Goldstone modes and the redundancy principle

To investigate SSB and the appearance of GMs in the spin-1 biquadratic model, it is convenient to introduce the Cartan generators $H_{1}$ and $H_{2}$, with the raising operators $E_{1}$, $E_{2}$, $E_{3}$ and the lowering operators $F_{1}$, $F_{2}$, $F_{3}$. These obey the properties $[H_{1},E_{1}]=2E_{1}$, $[H_{1},F_{1}]=-2F_{1}$, $[E_{1},F_{1}]=H_{1}$, $[H_{2},E_{2}]=2E_{2}$, $[H_{2},F_{2}]=-2F_{2}$, $[E_{2},F_{2}]=H_{2}$, $[H_{1}-H_{2},E_{3}]=2E_{3}$, $[H_{1}-H_{2},F_{3}]=-2F_{3}$ and $[E_{3},F_{3}]=H_{1}-H_{2}$, together with $[F_{1},F_{2}]=0$, $[F_{2},F_{3}]=0$, $[H_{1},F_{2}]=-F_{2}$ and $[H_{2},F_{1}]=-F_{1}$. Here, $H_{1}$, $E_{1}$, $F_{1}$, $H_{2}$, $E_{2}$, $F_{2}$, $E_{3}$, and $F_{3}$ may be expressed in terms of $J_{\eta}$, $\eta=1,\ldots,8$, as presented in Sec. A of the Supplementary Material.

If the highest weight state $|\rm{hws}\rangle$ is chosen to be $|\rm{hws}\rangle=|1...1\rangle$, where $|1\rangle$, together with $|0\rangle$ and $|\rm{-}1\rangle$, are the eigenvectors of $S_{z,j}$, with the eigenvalues being $1$, [math] and $-1$, respectively, then the expectation values of the local components $H_{1,j}$ and $H_{2,j}$ for the Cartan generators $H_{1}$ and $H_{2}$ are given by $\langle 1|H_{1,j}|1\rangle=1$ and $\langle 1|H_{2,j}|1\rangle=1$, if $j$ is odd, and $\langle 1|H_{1,j}|1\rangle=0$ and $\langle 1|H_{2,j}|1\rangle=1$, if $j$ is even. It follows that for $E_{\alpha}$ and $F_{\alpha}$ ($\alpha=1$, $2$ and $3$), one may choose $F_{\alpha,j}$ and $E_{\alpha,j}$ as the interpolating fields nielsen , respectively. Given $\langle[E_{1},F_{1,j}]\rangle\propto\langle H_{1,j}\rangle$, $\langle[E_{1,j},F_{1}]\rangle\propto\langle H_{1,j}\rangle$, $\langle[E_{2},F_{2,j}]\rangle\propto\langle H_{2,j}\rangle$, $\langle[E_{2,j},F_{2}]\rangle\propto\langle H_{2,j}\rangle$, $\langle[E_{3},F_{3,j}]\rangle\propto\langle H_{1,j}-H_{2,j}\rangle$ and $\langle[E_{3,j},F_{3}]\rangle\propto\langle H_{1,j}-H_{2,j}\rangle$, $E_{\alpha}$ and $F_{\alpha}$ are spontaneously broken, with $\langle H_{1,j}\rangle$ and $\langle H_{2,j}\rangle$ being the local order parameters. That is, six generators are spontaneously broken, but there are only two (linearly) independent local order parameters. This indicates that there are two type-B GMs, and so $N_{B}=2$. As a result of the Mermin-Wagner-Coleman theorem mwc1 ; mwc2 , no type-A GM survives in one spatial dimension, thus the number of type-A GMs $N_{A}$ must be 0. Hence, an apparent violation to the counting rule (1) occurs for the SSB pattern from $\rm{SU(3)}$ to $\rm{U(1)}\times\rm{U(1)}$. We remark that such a violation to the counting rule also occurs for SSB with type-A GMs typeared . A well-known example is phonons in three spatial dimensions, which break both the translational symmetries and the rotational symmetries, resulting in six broken generators, with only three type-A GMs.

This justifies the necessity to introduce a redundancy principle, which has to be valid for both type-A and type-B GMs. For our purpose, the redundancy principle may be stated as follows. The number of GMs should not exceed the maximum number of (linearly) independent local order parameters. As a result of the structure of Lie groups, this in turn implies that the number of type-B GMs should not exceed the maximum number of the commuting generators for the symmetry group, since an interpolating field is conjugate to the local component of its corresponding broken symmetry generator, which, together with a linear combination of the commuting generators, form a subgroup ${\rm SU(2)}$, as follows from the Watanabe-Brauner matrix brauner-watanabe . This is consistent with a mathematical theorem schafer . In particular, if the symmetry group is a semisimple Lie group, then the maximum number of the commuting generators becomes the rank $r$.

In our case, the rank $r$ is equal to 2 for the staggered symmetry group $\rm{SU(3)}$, then the number of type-B GMs must be equal to $N_{B}=2$, implying that two broken generators are redundant for the SSB pattern from $\rm{SU(3)}$ to ${\rm U(1)}\times{\rm U(1)}$.

IV Fibonacci-Lucas sequences and the golden spiral

The SSB pattern from $\rm{SU(3)}$ to ${\rm U(1)}\times{\rm U(1)}$ for the ferromagnetic spin-1 biquadratic model (3) does not fully account for the exponential ground state degeneracies with $L$ if only the highest weight state is considered. This is due to the fact that the dimension of the irreducible representation, generated from the highest weight state $|\rm{hws}\rangle$, for the staggered $\rm{SU}(3)$ is $(L+2)(3L+4)/8$ for $L$ even, and $(L+3)(3L+3)/8$ for $L$ odd. Physically, not all generators of the staggered symmetry group $\rm{SU}(3)$ commute with an additional $Z_{2}$ symmetry operation, which is denoted as $\sigma$, the one-site translation operation $T_{1}$ under PBCs or the bond-centred inversion operation $I_{b}$ with $L$ even, and the site-centred inversion operation $I_{s}$ with $L$ odd, under OBCs. Indeed, this is also reflected in the fact that the highest weight state $|1...1\rangle$ is the only one-site translation-invariant ground state under PBCs.

It is the presence of such an additional discrete $Z_{2}$ symmetry operation $\sigma$ that makes the ground state degeneracies exponential instead of being polynomial in $L$. Indeed, the $Z_{2}$ symmetry operation $\sigma$ intertwines with the lowering operators $F_{1}$ and $F_{2}$, and they repeatedly act on the highest weight state $|1...1\rangle$ and the generalized highest weight states, thus yielding the entire ground state subspace (cf. Sec. B of the Supplementary Material). This explains why the residual entropy is non-zero spins . In other words, a generalized highest weight state becomes indispensable, in addition to the highest weight state, to account for the exponential ground state degeneracies with $L$. Formally, the $m$-th generalized highest weight state $|{\rm hws}\rangle_{g}^{m}$ is defined recursively as $|{\rm hws}\rangle_{g}^{m}\notin V_{0}\oplus V_{1}\oplus...V_{m-1}$ and $E_{1}|{\rm hws}\rangle_{g}^{m}=0\mod(V_{0}\oplus V_{1}\oplus...V_{m-1})$, and $E_{2}|{\rm hws}\rangle_{g}^{m}=0\mod(V_{0}\oplus V_{1}\oplus...V_{m-1})$, where $V_{0}$ denotes the subspace spanned by the degenerate ground states generated from the highest weight states, whereas $V_{\mu}$ denotes the subspace spanned by the degenerate ground states generated from the $\mu$-th generalized highest weight state, where $\mu=1,2,\ldots,m$. Even though $|{\rm hws}\rangle_{g}^{m}$ are linearly independent to the states in the subspace $V_{0}\oplus V_{1}\oplus\ldots V_{m-1}$, all the states $E_{1}^{M_{1}}E_{2}^{M_{2}}|{\rm hws}\rangle_{g}^{m}$ ($M_{1}\geq 0$, $M_{2}\geq 0$ and $M_{1}+M_{2}\geq 1$) are linearly dependent to the states in the subspace $V_{0}\oplus V_{1}\oplus\ldots V_{m-1}$.

As an illustration, we show how to construct the degenerate ground states from the highest weight state and the generalized highest weight states for $L=2$, $4$ and $6$ in Sec. B of the Supplementary Material. The necessity to introduce a generalized highest weight state lies in the fact that only the degenerate ground states generated from the highest weight state and the generalized highest weight states are amenable to further analysis, which allows us to establish the scale invariance of the degenerate ground states, and to evaluate the entanglement entropy.

A remarkable fact is that there is a parity effect between even $L$ and odd $L$ under PBCs. This is seen from the observation that, instead of being staggered ${\rm SU(3)}$ for even $L$, the symmetry group reduces to uniform ${\rm SU(2)}$, generated from $S_{x}$, $S_{y}$ and $S_{z}$, for odd $L$, if PBCs are adopted. As a consequence, there is only one type-B GM, arising from the SSB pattern, namely ${\rm SU(2)}\rightarrow{\rm U(1)}$, if $L$ is odd under PBCs. This is in sharp contrast to two type-B GMs arising from the SSB pattern ${\rm SU(3)}\rightarrow{\rm U(1)}\times{\rm U(1)}$ if $L$ is even. However, in both cases, the presence of the generalized highest weight states ensures that the ground state degeneracies are exponential with $L$, regardless of $L$ being even or odd.

The exponential ground state degeneracies with $L$ are also relevant to the fact that a fractal structure underlies the highly degenerate ground states, as follows from the possibility for an exact singular value decomposition (see below) shiqianqian . Given that a wave function itself is not observable, it is necessary to choose a physical observable to reveal the scale invariance underlying such a fractal. Indeed, if the residual entropy $S_{\!r}$ is chosen as an observable, then the underlying self-similar geometric object turns out to be the golden spiral. That is, the ground state degeneracies ${\rm dim}(\Omega_{L}^{\rm OBC})$ and ${\rm dim}(\Omega_{L}^{\rm PBC})$ must be exponential with $L$, when $L\gg 1$, with $\Omega_{L}^{\rm OBC}$ and $\Omega_{L}^{\rm PBC}$ being the ground state subspaces under OBCs and under PBCs. Here we remark that an exponential function with an irrational base is involved, which must be less than $3$, but greater than $2$, since a simple estimate indicates that the ground state degeneracies are less than $3^{L}$, but greater than $2^{L}$. Given ${\rm dim}(\Omega_{L}^{\rm OBC})$ and ${\rm dim}(\Omega_{L}^{\rm PBC})$ are integer valued, we encounter a situation to express an integer in terms of an irrational number – the so-called Binet formula binet . A plausible surmise is that the ground state degeneracies satisfy a three-term recursive relation ${\rm dim}(\Omega_{L+2})=P\,{\rm dim}(\Omega_{L+1})-Q\,{\rm dim}(\Omega_{L})$ for $\Omega_{L}=\Omega_{L}^{\rm OBC}$ or $\Omega_{L}^{\rm PBC}$, with a characteristic equation being the quadratic equation $X^{2}-PX+Q=0$, where $P$ and $Q$ are co-prime integers: $(P,Q)=1$. In fact, we are led to $P=3$ and $Q=1$ (for the details, cf. Sec. C of the Supplementary Material). As a consequence, we have

[TABLE]

Here $R$ is the golden ratio $R=(\sqrt{5}-1)/2$. More precisely, the ground state degeneracies ${\rm dim}(\Omega_{L}^{\rm OBC})$ and ${\rm dim}(\Omega_{L}^{\rm PBC})$ constitute two Fibonacci-Lucas sequences. Hence, the residual entropy is $S_{\!r}=-2\log R$. This is consistent with the previous results spins . In fact, the three-term recursive relations for the two Fibonacci-Lucas sequences are identical to those for the ground state degeneracies spins ; saleur ; katsura ; moudgalya when $L\geq 2$ for OBCs and $L\geq 3$ for PBCs. Physically, the non-zero residual entropy $S_{\!r}$ measures the disorder arising from the intertwining nature of the combined action of the additional $Z_{2}$ symmetry operation $\sigma$, and the lowering operators $F_{1}$ and $F_{2}$ on the highest weight state as well as the generalized highest weight states (cf. Sec. B of the Supplementary Material).

V Degenerate ground states from the highest weight state and the generalized highest weight states

For the ferromagnetic spin-1 biquadratic model under consideration, sequences of the highly degenerate ground states are generated from the repeated action of $F_{1}$ and $F_{2}$, combining with the additional $Z_{2}$ symmetry operation $\sigma$ and the time-reversal operation $K$, on both the highest weight state and the generalized highest weight states.

Generically, a generalized highest weight state, with a period $p$ being an even integer, may be written as $|s_{1}s_{2}...s_{p}...s_{1}s_{2}...s_{p}\rangle$. Then, a sequence of the degenerate ground states, with the system size $L$ being a multiple of $p$, are generated from the repeated action of two of the three lowering operators $F_{1}$, $F_{2}$ and $F_{3}$ on such a generalized highest weight state $|s_{1}s_{2}...s_{p}...s_{1}s_{2}...s_{p}\rangle$, given $F_{3}=T_{1}F_{1}T_{1}$ under PBCs and $F_{3}=I_{b}F_{1}I_{b}$ under OBCs. We remark that the highest weight state itself $|1...1\rangle$ is exceptional, in the sense that it may be regarded as a special case, with the period $p$ being 1. As a result, the degenerate ground states generated from the highest weight states have periodicity $2$, due to the staggered structure of the lowering operators $F_{1}$, $F_{2}$ and $F_{3}$.

We may choose the two lowering operators $F_{1}$ and $F_{2}$ as an example. Hence, a sequence of the degenerate ground states, denoted as $|L,M_{1},M_{2}\rangle_{p}$, are generated from the repeated action of the two lowering operators $F_{1}$ and $F_{2}$ on $|s_{1}s_{2}...s_{p}...s_{1}s_{2}...s_{p}\rangle$:

[TABLE]

where $Z_{p}(L,M_{1},M_{2})$ is introduced to ensure that $|L,M_{1},M_{2}\rangle_{p}$ is normalized. We emphasize that $M_{1}$ and $M_{2}$ are subject to some constraints, which may be related to $L$.

Physically, the occurrence of the degenerate ground states with the period $p$ ($p=1,2,4,6,\ldots$) results from SSB of the one-site translation symmetry to a discrete symmetry group $\bigoplus_{p}\mathscr{Z}_{p}$, with $\mathscr{Z}_{p}$ being generated by the $p$-site translation under PBCs, that always accompanies SSB of the staggered symmetry group ${\rm SU(3)}$. It is this dependence of $p$ on $L$ that renders the ground state degeneracy with $L$ being exponential. Importantly, among the degenerate ground states, there is a translation-invariant ground state with $p=1$, dimerized ground states with $p=2$, tetramerized ground states with $p=4$, and so on, in sharp contrast to the antiferromagnetic spin-1 biquadratic model afflecksun1 , where only dimerized ground states are involved.

V.1 Degenerate ground states generated from the highest weight state $|1...1\rangle$

There are distinct choices for the highest weight state $|\rm{hws}\rangle$, two of which are as follows. One is the translation-invariant factorized ground state: $|\rm{hws}\rangle=|1...1\rangle$, the other is the factorized ground state without the one-site translational invariance: $|\rm{hws}\rangle=|0_{x}0_{y}...0_{x}0_{y}\rangle$, under PBCs. Here $|0_{x}\rangle$ and $|0_{y}\rangle$ are the eigenvectors of $S_{x,j}$ and $S_{y,j}$, with their respective eigenvalue being [math]. We remark that the two choices are unitarily equivalent under a local unitary transformation $U$, satisfying $K_{x}=UK_{y}U^{\dagger}$, $K_{y}=UK_{z}U^{\dagger}$ and $K_{z}=UK_{x}U^{\dagger}$, with $K_{x}=J_{5}$, $K_{y}=J_{6}$ and $K_{z}=J_{3}$ being the three generators of a symmetry subgroup ${\rm SU}(2)$. Here $U$ takes a product form $\prod_{j}U_{j}$, where $U_{j}=A+BS_{z,j}+CS_{z,j}^{2}+D[(S_{x,j})^{2}-(S_{y,j})^{2}+S_{x,j}S_{y,j}+S_{y,j}S_{x,j}]$, with $A=i$, $B=(-1)^{j}(-1+i)\sqrt{2}/4$, $C=(-1)^{j}\sqrt{2}/4+[(-1)^{j}\sqrt{2}-4]i/4$ and $D=(-1+i)\sqrt{2}/4$.

Note that other choices may be induced from the symmetric group ${\rm S}_{3}$, arising from the cyclic permutations with respect to $x$, $y$, and $z$. As a consequence, the entanglement entropy $S(n)$ for distinct sequences of the degenerate ground states, corresponding to the different choices for the highest weight state $|\rm{hws}\rangle$, must be identical. Therefore, we only need to focus on one choice: $|\rm{hws}\rangle=|1...1\rangle$.

A sequence of the degenerate ground states $|L,M_{1},M_{2}\rangle_{2}$ ($M_{1}=0$, …, $L/2$ and $M_{2}=0$, …, $L$) in a two-site periodic structure, are generated from the repeated action of the lowering operators $F_{1}$ and $F_{2}$ on the highest weight state $|1...1\rangle$:

[TABLE]

where $Z_{2}(L,M_{1},M_{2})$ is introduced to ensure that $|L,M_{1},M_{2}\rangle_{2}$ is normalized. It takes the form

[TABLE]

A derivation of the concrete expression for $Z_{2}(L,M_{1},M_{2})$ is presented in Sec. D of the Supplementary Material.

In fact, in addition to $|L,M_{1},M_{2}\rangle_{2}$ in Eq. (7), a sequence of the degenerate ground states $|L,M_{1},M_{3}\rangle_{2}$ ($M_{1}=0$, …, $L/2+M_{3}$ and $M_{3}=0$, …, $L/2$), are generated from the repeated action of the lowering operators $F_{1}$ and $F_{3}$ on the highest weight state $|1...1\rangle$:

[TABLE]

where $Z_{2}(L,M_{1},M_{3})$ is introduced to ensure that $|L,M_{1},M_{3}\rangle_{2}$ is normalized. It takes the form

[TABLE]

A derivation of the concrete expression for $Z_{2}(L,M_{1},M_{3})$ is presented in Sec. D of the Supplementary Material. The degenerate ground states, generated from the highest weight state $|\rm{hws}\rangle=|0_{x}0_{y}...0_{x}0_{y}\rangle$ are discussed in Sec. E of the Supplementary Material.

V.2 Degenerate ground states generated from the generalized highest weight states

The emergence of a bunch of the generalized highest weight states is not only responsible for the exponential ground state degeneracies with $L$, but also accounts for the difference between the ground state degeneracies under OBCs and PBCs (cf. Sec. B of the Supplementary Material).

For our purpose, we choose two typical generalized highest weight states: $|1110...1110\rangle$ and $|10\rm{-}1\rm{-}101...10\rm{-}1\rm{-}101\rangle$, which exhibit a four-site and six-site periodic structure, i.e., with $p=4$ and $p=6$.

A sequence of the degenerate ground states $|L,M_{2},M_{3}\rangle_{4}$ ($M_{2}=0,\ldots,3L/4$ and $M_{3}=0,\ldots,L/4$), in a four-site periodic structure, are generated from the repeated action of the lowering operators $F_{2}$ and $F_{3}$ on $|1110...1110\rangle$, with the period $p$ being four:

[TABLE]

where $Z_{4}(L,M_{2},M_{3})$ is introduced to ensure that $|L,M_{2},M_{3}\rangle_{4}$ is normalized. It takes the form

[TABLE]

A derivation of $Z_{4}(L,M_{2},M_{3})$ is presented in Sec. D of the Supplementary Material.

In the six-site periodic structure, a sequence of the degenerate ground states $|L,M_{1},M_{2}\rangle_{6}$ ($M_{1}=0$, …, $L/3$ and $M_{2}=0$, …, $L/3$) are generated from the repeated action of the lowering operators $F_{1}$ and $F_{2}$ on $|10\rm{-}1\rm{-}101...10\rm{-}1\rm{-}101\rangle$, with the period $p$ being six:

[TABLE]

where $Z_{6}(L,M_{1},M_{2})$ is introduced to ensure that $|L,M_{1},M_{2}\rangle_{6}$ is normalized. It takes the form

[TABLE]

A derivation of $Z_{6}(L,M_{1},M_{2})$ is presented in Sec. D of the Supplementary Material.

VI Singular value decomposition, entanglement entropy and fractal dimension

Since the ground state subspace is spanned by the linearly independent (degenerate) ground states generated from the action of lowering operators $F_{1}$ and $F_{2}$, combining with the additional $Z_{2}$ symmetry operation $\sigma$ and the time-reversal operation $K$, on the highest weight state as well as the generalized highest weight states, we only need to consider the singular value decomposition for such a degenerate ground state. We are able to show that such a generic degenerate ground state admits an exact singular value decomposition, thus implying the self-similarities underlying the degenerate ground states as a whole. However, it is presumably a formidable task to evaluate the norm for such a degenerate ground state. Therefore, we restrict ourselves to two typical generalized highest weight states, with the period $p$ being four and six, respectively, in addition to the highest weight state.

The degenerate ground states $|L,M_{1},M_{2}\rangle_{p}$, with $p$ being an even number, in Eq. (6) admit an exact singular value decomposition, with

[TABLE]

where $n$ is even, and the singular values $\lambda(L,n,k_{1},k_{2},M_{1},M_{2})$ take the form

[TABLE]

Here $Z_{p}(n,k_{1},k_{2})$ and $Z_{p}(L-n,M_{1}-k_{1},M_{2}-k_{2})$ take the same form as $Z_{p}(L,M_{1},M_{2})$.

For the degenerate ground states $|L,M_{1},M_{2}\rangle_{p}$, the entanglement entropy $S_{\!L}(n,M_{1},M_{2})$ follows from

[TABLE]

where $\Lambda(L,n,k_{1},k_{2},M_{1},M_{2})$ are the eigenvalues of the reduced density matrix $\rho_{L}(n,M_{1},M_{2})$, given by $\Lambda(L,n,k_{1},k_{2},M_{1},M_{2})=[\lambda(L,n,k_{1},k_{2},M_{1},M_{2})]^{2}$.

For fixed fillings $f_{1}=M_{1}/L$ and $f_{2}=M_{2}/L$ in the thermodynamic limit $L\rightarrow\infty$, the entanglement entropy $S_{\!L}(n,M_{1},M_{2})$ becomes $S(n,f_{1},f_{2})$. The scaling of the entanglement entropy $S(n,f_{1},f_{2})$ with the block size $n$, in the thermodynamic limit, for any non-zero fillings $f_{1}$ and $f_{2}$, takes the same form as Eq. (2). This is consistent with a generic but heuristic argument in Ref. shiqianqian . Accordingly, the fractal dimension $d_{f}$ is identical to the number of type-B GMs, with $d_{f}=N_{B}$ where $N_{B}=2$.

Similarly, a singular value decomposition may be performed for the degenerate ground states $|L,M_{1},M_{3}\rangle_{2}$, $|L,M_{2},M_{3}\rangle_{4}$ and $|L,M_{1},M_{2}\rangle_{6}$ given in equations (9), (11) and (13) (cf. Sec. F of the Supplementary Material). We shall see that the entanglement entropy scales with block size $n$ for each of the three choices in the same way as Eq. (2), with $n$ being a multiple of two, a multiple of four and a multiple of six, respectively.

The emergence of the logarithmic scaling behaviour of the entanglement entropy with the block size $n$, when the system size $L$ approaches the thermodynamic limit, is demonstrated in Sec. G of the Supplementary Material.

Plots of the entanglement entropy against the block size in the thermodynamic limit are shown in Fig. 1 for different filling factors. The results imply that the contribution from the filling factors is a non-universal additive constant as in Eq. (2).

VII summary

We have shown that the physical and mathematical structure underlying the highly degenerate ground states of the ferromagnetic spin-1 biquadratic model is particularly rich. For this model the SSB pattern from $\rm{SU(3)}$ to $\rm{U(1)}\times\rm{U(1)}$, with two type-B GMs, leads to scale, but not conformally, invariant degenerate ground states. This indicates that an abstract fractal emerges in the Hilbert space, with the fractal dimension being identical to the number of type-B GMs. As such, the presence of such a quantum state challenges the folklore that scale invariance implies conformal invariance polyakov , with its ramifications for our understanding of quantum critical phenomena remaining to be clarified fm .

We found it necessary to introduce a redundancy principle, to keep consistency with the counting rule (1) for type-B GMs, as also happens for type-A GMs. Given the staggered nature of the symmetry group ${\rm SU}(3)$, the ground state degeneracies are exponential with the system size $L$, which accounts for the fact that the residual entropy is non-zero. This may be explained based on an observation that the degenerate ground states exhibit a periodic structure, which in turn results from SSB of the one-site translation symmetry to the discrete symmetry group $\bigoplus_{p}\mathscr{Z}_{p}$ that always accompanies SSB of staggered ${\rm SU(3)}$. Remarkably, the ground state degeneracies constitute the two Fibonacci-Lucas sequences, thus providing an unexpected connection to an ancient mathematical gem. As a consequence, the ground state degeneracies are asymptotically the golden spiral, a well known self-similar geometric object. In addition, sequences of degenerate ground states generated from highest and generalized highest weight states have been constructed in Section V to establish that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit, with the prefactor being half the number of type-B Goldstone modes.

VIII Acknowledgements

We thank John Fjaerestad for enlightening discussions and Sanjay Moudgalya and Hosho Katsura for bringing their work to our attention.