Entropy, gap and a multi-parameter deformation of the Fredkin spin chain
EEntropy, gap and a multi-parameter deformation of the Fredkinspin chain
Zhao Zhang and Israel KlichDepartment of Physics, University of Virginia, Charlottesville, 22904, VA
We introduce a multi-parameter deformation of the Fredkin spin 1 / t i along the chain. The parameters are introduced in such a way to maintainthe system frustration-free while allowing to explore a range of possible phases. Inthe case where the parameters are uniform, and a color degree of freedom is addedwe establish a phase diagram with a transition between an area law and a volumelow. The volume entropy obtained for half a chain is n log s where n is the half-chainlength and s is the number of colors. Next, we prove an upper bound on the spectralgap of the t > , s > O ((4 s ) n t − n / ), similar to a recent aresult about the deformed Motzkin model, albeit derived in a different way. Finally,using an additional variational argument we prove an exponential lower bound onthe gap of the model for t > , s = 1, which provides an example of a system withbounded entanglement entropy and a vanishing spectral gap. I. INTRODUCTION
Entanglement is one of the central quantum phenomena that distinguish quantum systemsfrom their classical counterparts. It has profound implications in many different contextsof modern physics and has both driven theoretical discussions of the foundations of quan-tum mechanics and motivated promising applications in quantum information and quantumcomputation. In quantum many-body physics, the study of entanglement is focused on thesurvival, or lack thereof, of entanglement between individual particles when a large numberof particles organize themselves with interactions in a condensed matter system [1]. Someof the characteristics of the correlations generated this way are quantified by the concept of‘entanglement entropy’ and it’s scaling with the system size.Scaling of entanglement entropy is often closely related to the spectral gap of a system, a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r although an exact relation is still somewhat elusive, especially in higher dimensions. In agapped system, correlations are short-ranged, and entanglement entropy is expected to obeyan “area law”, which says that the entanglement entropy of a region scales with the area ofthe boundary of the region as opposed to its volume. In one dimension the entanglemententropy of a subsystem is bounded by a constant regardless of its length, as shown in[3, 4]. For gapless systems, (1+1)-dimensional conformal field theory studies find logarithmicscaling [5–7]. Fermi liquids exhibit a logarithmic violation of area law behavior scaling inany dimension [8, 9], and require control of multidimensional dimensional gineralizations ofthe Szego limit theorems as described by Widom’s conjecture [9–11].Recently, efforts have been made to find one-dimensional spin systems with more severeviolations of the area law [12–15]. These exotic scalings are achieved at the price of eitherintroducing a high-dimensional local Hilbert space or sacrificing translational invariance.Based on previous work by Bravyi et al [16], Movassagh and Shor [17] first introduced ahighly-entangled (power-law scaling) integer spin- s chain ( s >
1) that enjoys several physi-cally appealing features, including relatively small dimensionality of the local Hilbert space,short-range interaction and translational invariance. Furthermore, the model Hamiltonian isfrustration free: it can be written as a sum of local projectors sharing a unique simultaneousground state. Inspired by this work, in [18] we have used the idea to describe a class ofHamiltonians with a tunable parameter t that exhibits a novel quantum phase transition.The t > L = 2 n . An initial estimate of exponential decay has been improved by Levineand Movassagh [19] to an unusually fast decay as exp( − n / t < t > s = 1 and t >
1. We explain thisbehavior by establishing a variational excited state that has vanishing energy t > , s = 1in the thermodynamic limit.The paper is structured as follows. In Section II, we briefly review the constituting ingre-dients Dyck walk and Fredkin interaction of the Hamiltonian and ground state of Fredkinspin chain and show how the deformation parameter t fit in without compromising its frus-tration free feature of the Hamiltonian and the uniqueness of its ground state. In Section III,we illuminate the mechanism behind the linear scaling of entanglement entropy of Fredkinspin chain by introducing the non-commutative ‘height’ and ‘shift’ operators, without re-sorting to any mathematical knowledge beyond analysis. In Section IV, we exploit featuresof the Hamiltonian and its ground state to construct low energy excitation states for the s > s = 1 models that have exponentially small gaps in the t > II. HAMILTONIAN AND GROUND STATE
We start with a brief review of Dyck walks, which span the ground state subspace of theHilbert space of the Fredkin spin chain Hamiltonian introduced in [21, 22].
Definition 1.
A Dyck walk (or path) on n steps is any path from (0 , to (0 , n ) with steps (1 , and (1 , − that never passes below the x -axis. A Dyck walk can be mapped to a spin configuration of a spin-1 / | σ σ . . . σ n (cid:105) with σ k = +1 / − / ,
1) or (1 , − k th step respectively. When each step isassigned a color c k from a palette of s colors, | ↑ ck (cid:105) ( | ↓ ck (cid:105) ) corresponds to a state σ k = c k / σ k = − c k / t that deforms the Fredkin Hamiltonian of [21, 22] whileremaining frustration free. The Hamiltonian is given by: H ( s, t ) = H F ( s, t ) + H X ( s ) + H ∂ ( s ) , (2.1)where H F ( s, t ) = n − (cid:88) j =2 s (cid:88) c ,c ,c =1 (cid:0) | φ c ,c ,c j,A (cid:105)(cid:104) φ c ,c ,c j,A | + | φ c ,c ,c j,B (cid:105)(cid:104) φ c ,c ,c j,B | (cid:1) (2.2) H X ( s ) = (cid:80) n − j =1 [ (cid:80) c (cid:54) = c | ↑ c j ↓ c j +1 (cid:105)(cid:104)↑ c j ↓ c j +1 | (2.3)+ (cid:80) sc ,c ( | ↑ c j ↓ c j +1 (cid:105) − | ↑ c j ↓ c j +1 (cid:105) )( (cid:104)↑ c j ↓ c j +1 | − (cid:104)↑ c j ↓ c j +1 | )] , (2.4)and H ∂ ( s ) = (cid:80) sc =1 ( | ↓ c (cid:105)(cid:104)↓ c | + | ↑ c n (cid:105)(cid:104)↑ c n | ) . (2.5)The projectors in H F are defined using: | φ c ,c ,c j,A (cid:105) = 1 (cid:112) | t A,j | ( (cid:12)(cid:12) ↑ c j ↑ c j +1 ↓ c j +2 (cid:11) − t A,j (cid:12)(cid:12) ↑ c j ↓ c j +1 ↑ c j +2 (cid:11) ) (2.6) | φ c ,c ,c j,B (cid:105) = 1 (cid:112) | t B,j | ( (cid:12)(cid:12) ↑ c j ↓ c j +1 ↓ c j +2 (cid:11) − t B,j (cid:12)(cid:12) ↓ c j ↑ c j +1 ↓ c j +2 (cid:11) ) (2.7) j j + 1 t j − A ( ) − = t jB ( ) − FIG. 1: Different ways of “flattening” a hill must have the same amplitude. Note that the colorshave been interchanged during the procedure, this, however, is not an obstruction as all coloringsappear with the same amplitude. with the condition that t Bj = t Aj − .The Fredkin gate projectors in H F allows a pair of ↑↓ neighboring spins (with the samecolor enforced by the first term in H X ) to move freely around its left or right third neighborand still appear in the ground state superposition, but now with a different probabilityamplitude. The second term in H X ensures that otherwise identical Dyck paths with differentcoloring have the same weight. And the boundary term H ∂ (together with the Fredkinprojectors) penalizes paths that go below 0 at any point along the chain. Notice thatanalogous to [18], the simplest choice is a parameter t = t A = t B being the same in the twoprojectors of H F is the one employed in [23, 24], but is only a subset of the parameter spacethat leaves the Hamiltonian frustration free. More generally, we introduce parameters t A and t B , for the two projectors in H F . Then any set of { t Aj , t Bj } that satisfies the condition t Aj = t Bj +1 for all j ’s would guarantee the Hamiltonian to be frustration free . The pointis illustrated in Fig. 1. In particular, we may specify a Hamiltonian with a frustration-freeground state by picking any set of the t Aj parameters.We now want to characterize the ground state of the system. First, let us denote h ( l ) tobe the height of the Dyck path after step l , that is, for a spin configuration | w (cid:105) describinga Dyck path, s (cid:88) c =1 l (cid:88) j =1 σ zj,c | w (cid:105) = h ( l ) | w (cid:105) , (2.8) Note that there is no parallel condition t Bj = t Aj +1 . h(1) h(2) h(3) h(4) h(5) h(6) h(7) h(8) h(0) FIG. 2: A spin configuration corresponding to a colored Dyck path and the corresponding heightfunction h ( l ). where σ zj,c is the Pauli matrix giving ± j is in state ↑ cj or ↓ cj , respectively. The heightfunction is illustrated for a generic Dyck path in Fig. 2. To find the relative amplitudeof this spin configuration as compared with the lowest possible spin configuration, we usesuccessively the Fredkin moves to | ↑↑↓(cid:105) −→ t A | ↑↓↑(cid:105) to ”flatten” the hill. The process isdescribed in Fig. 3. In this way, the weight of each Dyck paths is related to the weightof the lowest height, | ↑↓↑↓ ... ↑↓(cid:105) path. Note that we have suppressed the color index inthis treatment since, as mentioned above, in the ground state superposition all admissiblecolorings should appear with the same amplitude.The amplitude of a given Dyck path in the ground state of the model is thus given bycounting the number of ”diamonds” associated with each t Aj , and can be written in the form: | GS (cid:105) = 1 N (cid:88) w ∈{ s − coloredDyck walks } e (cid:80) n − l =2 [ h ( l )2 ] log( t Al − ) | w (cid:105) , (2.9)where [ x ] is the integer part of x and N is a normalization factor. In the case where t Aj ≡ t ,the ground state can be simply related to the area under the path as: | GS (cid:105) = 1 N (cid:48) (cid:88) w ∈{ s − coloredDyck walks } t A ( w ) | w (cid:105) . (2.10) h(1) h(2) h(3) h(4) h(5) h(6) h(7) h(0) t A,1 t A,1 t A,4 t A,4 t A,3 t A,1 t A,4 t A,3 t A,1 t A,5 h(8)
FIG. 3: The “flattening” of a hill and it’s amplitude. Starting from the left, we reduce the first byusing the Fredkin move | ↑↑↓(cid:105) −→ t A | ↑↓↑(cid:105) . This process is repeated for each peak until the lowestheight hill is achieved. III. ENTANGLEMENT ENTROPY
In this section, we employ the simplest choice of parameters which is translational in-variant t A = t B = t everywhere. When t = 1, the entanglement entropy of the ground statescales as log n for s = 1 and as √ n for s > ↑↓ pair, it is separated from its own partner paired in the same color,which is the first unpaired down spin to its right (or up spin to its left). This way, whena pair of spins required to be in the same color are shifted to different subsystems of thechain, they become a source of entanglement entropy between the two subsystems. Tuningthe parameter t to favor higher paths in the ground state superposition will now enhancethe more substantial contribution from those with more unpaired spins in one subsystem.To put this in a mathematical way, we decompose the ground state into tensor products ofstates in the left and right halves of the chain. | GS (cid:105) = n (cid:88) m =0 √ p n,m (cid:88) x ∈{↑ , ↑ ,..., ↑ s } m | ˆ C ,m,x (cid:105) ,...,n ⊗ | ˆ C m, , ¯ x (cid:105) n +1 ,..., n , (3.1)where | ˆ C p,q,x (cid:105) is a weighted superposition of spin configurations with p excess ↓ , q excess ↑ and a particular coloring x of the unmatched arrows, such that (cid:104) GS | ( | ˆ C ,m,x (cid:105) ,...,n ⊗| ˆ C m, , ¯ x (cid:105) n +1 ,..., n ) (cid:54) = 0, and ¯ x is the coloring in the second half of the chain that matches x . The decomposition gives the Schmidt number p n,m ( s, t ) = M n,m ( s, t ) N n ( s, t ) , (3.2)where M n,m ( s, t ) ≡ s n − m (cid:88) w ∈{ st half of Dyckwalks stopping at ( n,m ) } t A ( w ) , (3.3) N n ( s, t ) ≡ n (cid:88) m =0 s m M n,m ( s, t ) . (3.4)And the entanglement entropy of the half chain in the ground state is given by S n ( s, t ) = − n (cid:88) m =0 s m p n,m ( s, t ) log p n,m ( s, t ) . (3.5)To study the behavior of M n,m as a function of m , we observe that they satisfy thefollowing recurrence relations, M k +1 ,k +1 = t k + M k,k ,M k +1 ,m = st m + M k,m +1 + t m − M k,m − , for 0 < m < k,M k +1 , = st M k, . (3.6)Notice that the M n,m is only non-vanishing for m ’s of same parity as n .From these relations, we can see that for large enough t , M n,m will be monotonicallyincreasing as we increase m by increments of 2. Paths with height in the middle scaling as O ( n ) will contribute more to the entanglement entropy from the s O ( n ) possible colorings ofunmatched spins. In particular, the half chain entanglement entropy will also scale linearlywith system size n . In the next subsection, we give a rigorous proof that this is true in thethermodynamic, and that this critical phase of large entanglement spans the entire half line t > A. t > , s > phase: Volume scaling of entropy. In this section we repeat the steps taken in [18], to prove volume scaling for weightedMotzkin walks, with a few modifications. For arbitrary t >
1, the non-zero entries of M n,m are not necessarily monotonic in terms of m , but we can still show that for a given n , M n,m reaches its maximum at some m = m ∗ , within a finite distance away from m = n independent of the system size n itself. This is not obvious in the step-by-step recurrencerelations, but becomes clear as we take into account the accumulated effect of the evolutionof the coefficients M n,m with respect to n . To see this, we summarize (3.6) in the followingoperator formalism.As in [ ? ], we represent the distributions of M k,m as components of the state at ‘time’ k during the ‘evolution’ in a basis spanned by | m (cid:105) , m = 0 , , , . . . . |M k (cid:105) = ∞ (cid:88) m =0 M k,m | m (cid:105) , M k,m = 0 if m > k. (3.7)We we define ‘shift’ and ‘height’ operators to describe the ‘evolution’ of the the states |M k (cid:105) as S| m (cid:105) = | m − (cid:105) , S| (cid:105) = 0; (3.8) H| m (cid:105) = m | m (cid:105) . (3.9)One can check that the recurrence relations (3.6) translate to (cid:104) m |M k +1 (cid:105) = M k +1 ,m = st m + (cid:104) m + 1 |M k (cid:105) + t m − (cid:104) m − |M k (cid:105) = (cid:104) m | st H + S + t H− S † |M k (cid:105) , which gives us: |M k +1 (cid:105) = t H ( s √ t S + 1 √ t S † ) |M k (cid:105) . (3.10)Using the commutation relation t k H ( s √ t S + 1 √ t S † ) = ( st − ( k − ) S + t k − S † ) t k H , (3.11)0we keep moving the t k H operators all the way to the right until it disappears when actingon | (cid:105) we obtain: |M n (cid:105) = [ t H ( s √ t S + 1 √ t S † )] n |M (cid:105) = (cid:126) K n (cid:89) k =1 ( st − ( k − ) S + t k − S † ) | (cid:105) . (3.12)Here (cid:126) K denotes ordering the multiplications in the product such that factors with greater k value are on the right. For t > S † term for large k . In other words, at some point during the evolution, the distribution of M m starts shifting at velocity 1 to the right along the m axis without much spreading. Fora larger t , this happens shortly after the evolution starts, while for smaller values of t , ittakes longer to reach this stable propagation. In any case, as we show below, the maximumof M n,m is a within finite distance away from m = n . Lemma 1.
Let m ∗ be such that sup m M n,m = M n,m ∗ , then ∃ N < n , such that when t > , m ∗ ∈ [ n − N , n ] .Proof. Let |M (cid:48) n (cid:105) = (cid:126) K n (cid:89) k = N +1 ( st − ( k − ) S + t k − S † ) | (cid:105) . (3.13)Note that t − ( k − ) (cid:107) st − ( k − ) S(cid:107) ≤ st − k − ) ≡ c k , (3.14)so that: (cid:107) st − ( k − ) S + t k − S † (cid:107) < t k − (1 + c k )we thus have (cid:107) t − (cid:80) nk = N ( k − ) |M (cid:48) n (cid:105) − | n − N (cid:105)(cid:107) ≤ (cid:81) nk = N +1 ( c k + 1) − < e (cid:80) ∞ k = N c k − e stt − t − N − ≡ f ( s, t ) t − N − . The first inequality on the left follows from noting that | n − N (cid:105) appears in t − (cid:80) nk = N ( k − ) |M (cid:48) n (cid:105) with coefficient 1, and is exactly canceled. We have also used that1 x + 1 ≤ e x . Next, (cid:107) t − (cid:80) nk =1 ( k − ) |M n (cid:105) − (cid:126) K N (cid:89) k =1 ( st − k − ) S + S † ) | n − N (cid:105)(cid:107) ≤(cid:107) (cid:126) K N (cid:89) k =1 ( st − k − ) S + S † ) (cid:107) (cid:107) t − (cid:80) nk = N ( k − ) |M (cid:48) n (cid:105) − | n − N (cid:105)(cid:107) < ( f ( s, t ) t − N − N (cid:89) k =1 (1 + c k ) < ( f ( s, t ) t − N − e (cid:80) N k =1 c k < ( f ( s, t ) t − N − f ( s, t ) . Let M (cid:48) n,m = (cid:104) m | (cid:126) K N (cid:89) k =1 ( st − k − ) S + S † ) | n − N (cid:105) , (3.15)then clearly M (cid:48) n,m = 0 for m < n − N . If we choose N = f ( s, t ) < √ , − log log( f − s,t )+1)log f ( s,t ) log t , otherwise , (3.16)then (cid:107) t − n |M n (cid:105) − n (cid:88) m = n − N M (cid:48) n,m | m (cid:105)(cid:107) < M (cid:48) n,n ≤ sup m M (cid:48) n,m . (3.17)Therefore ∃ m ∗ ∈ [ n − N , n ], such that M n,m ∗ ≥ M n,m for all m .This allows us to prove the linear scaling of the entanglement entropy. Theorem 1.
In the state (2.10) , when t > , the entanglement entropy of half of the chainis bounded from below by S n > n log s + C ( s, t ) , where C ( s, t ) is an n independent constant.Proof. We separate a linear term from S n as follows (below we supress the n index in M n,m ): S n = (cid:80) nm =0 s m p m log s m − (cid:80) nm =0 s m p m log( s m p m ) > (cid:80) nm =0 s m p m m log s = (cid:80) nl =0 s n − l p n − l ( n − l ) log s = n log s − log s (cid:80) nl =0 s n − l M n − l (cid:80) nm (cid:48) =0 s m (cid:48) M m (cid:48) l (3.18)Taking m ∗ such that sup m M n,m = M n,m ∗ and using lemma 1, we see that n (cid:88) l =0 s n − l M m ∗ (cid:80) nm (cid:48) =0 s m (cid:48) M m (cid:48) l < n (cid:88) l =0 s n − l M m ∗ s m ∗ M m ∗ l = s n − m ∗ n (cid:88) l =0 s − l l< s N n (cid:88) l =0 s − l l < s N ∞ (cid:88) l =0 s − l l = s N +1 ( s − . Therefore, the remainder term on the right hand side of (3.18) is bounded.2One can see from the proof that the factor of s m is already enough to make the scalingof entropy linear, and all that is required for p m is that it doesn’t destroy this exponentialdependence on m . B. t < and any s : Bounded entanglement entropy. Contrary to the case studied above, when t <
1, we expect Dyck paths with smallerareas below to be exponentially preferred in the ground state superposition. But this time,for the entropy to reflect the predominance of lower path, where less mutual informationbetween the two subsystems can be stored, the behavior of p m needs to not only be decreasingexponentially with m , but also fast enough to overcome the exponential increasing s m factor.Considering that, we define˜ M n,m = s m M n,m , ˜ p n,m = ˜ M n,m (cid:80) nm =0 ˜ M n,m . (3.19)Substitution into (3.6) gives the following relations,˜ M k +1 ,k +1 = √ st k + ˜ M k,k , ˜ M k +1 ,m = √ s ( t m + ˜ M k,m +1 + t m − ˜ M k,m − ) , < m < k, ˜ M k +1 , = √ st ˜ M k, (3.20)To prove the entropy is bounded, we need the following lemmas. Lemma 2. ˜ M n +2 ,m > ˜ M n,m . (3.21) Proof.
From (3.12), we have |M n +2 (cid:105) = (cid:126) K n (cid:89) k =1 ( st − ( k − ) S + t k − S † )( st − ( n + ) S + t n + S † )( st − ( n +1+ ) S + t n +1+ S † ) | (cid:105) , = (cid:126) K n (cid:89) k =1 ( st − ( k − ) S + t k − S † )[ s t − n +1) S + s ( t + 1 t ) + t n +1) S † ] | (cid:105) = s ( t + 1 t ) |M n (cid:105) + (cid:126) K n (cid:89) k =1 ( st − ( k − ) S + t k − S † )[ s t − n +1) S + t n +1) S † ] | (cid:105) . | m (cid:105) ,with m = 0 , , . . . , n + 2, and we have: M n +2 ,m > M n,m , ˜ M n +2 ,m > ˜ M n,m ∀ m ≥ , n ≥ . Next we establish the following bound on ˜ p n,m : Lemma 3. ˜ p n,m < s t t m . (3.22) Proof.
By definition of ˜ p n,m , and using the recursion relation (3.20) twice consequtively,˜ p n,m = {√ s [ t m + √ s ( t m + ˜ M n − ,m +2 + t m + ˜ M n − ,m ) + t m − √ s ( t m − ˜ M n − ,m + t m − ˜ M n − ,m − )] } (cid:80) nm =0 ˜ M n,m = s t m [ t ˜ M n − ,m +2 + ( t + t ) ˜ M n − ,m + t − ˜ M n − ,m − ] (cid:80) nm =0 ˜ M n,m ≤ s t m [3 max { t ˜ M n − ,m +2 , ( t + t ) ˜ M n − ,m , t − ˜ M n − ,m − } ] (cid:80) nm =0 ˜ M n,m ≤ s t t m max { ˜ M n,m +2 , ˜ M n,m , ˜ M n,m − } (cid:80) nm =0 ˜ M n,m < s t t m . Lemma 2 was used in the last line.We now have the ingredients to prove the boundedness of entropy.
Theorem 2.
When < t < , s ≥ , there exists a constant C ( s, t ) independent of thesystem size n , that for any n , S n < C ( s, t ) .Proof. Using Lemma 3 we see that when m > m ≡ (cid:104) log( e t s )4 log t (cid:105) + 1 , (3.23)we have ˜ p n,m < s t t m < e . (3.24)4It is easy to check that the function − x log( x ) is monotonically increasing when x ∈ (0 , e ),in other words, for m > m ,˜ p n,m < s t t m < e = ⇒ − ˜ p n,m log ˜ p n,m < − s t t m (cid:0) log( 36 s t ) + 4 m log t (cid:1) . (3.25)Therefore S n = − n (cid:88) m =0 ˜ p n,m log ˜ p n,m + log s n (cid:88) m =0 ˜ p n,m m< − m (cid:88) m =0 ˜ p n,m log ˜ p n,m − ∞ (cid:88) m = m +1 s t t m (cid:0) log( 36 s t ) + 4 m log t (cid:1) + log s ∞ (cid:88) m =0 s t t m m< m + 1 e − s t m +2 − t log( 36 s t ) − s t m +2 ( m (1 − t ) + 1)( t − log t + 36 s t ( t − log s ≡ C ( s, t ) , where we used sup x ∈ (0 , − xlog ( x ) = e − for entropy terms with m ≤ m in the last inequal-ity.Notice our proof here does not rely on the fact that s >
1, and it applies to the s = 1case as well. IV. SCALING OF THE SPECTRAL GAPA. Super-exponential Upper bound in the t > , s = 1 Phase
Since entanglement entropy is a measure of correlation in the system, a high entangle-ment entropy indicates that the system is highly correlated and also a gapless spectrum (inthe thermodynamic limit) [3, 4]. As our model at t > , s > t = 1. Here, we give variational proof that the spectral gap for t > , s > Definition 2.
A prime Dyck walk is a Dyck walk that is always above the x-axis, except atthe endpoints.
By this definition, a Dyck walk is either prime or a concatenation of prime walks (Fig. 4exhibits a Dyck walk in solid line made of two prime walks and one in dashed line made ofthree prime walks).To construct a low energy variational excited state, we start with an auxiliary state thatprojects out all the walks in the ground state superposition whose longest prime walk has alength smaller than n + 1. That is, define: P n,> = { s-colored walks containing a prime walk of length l > n } , (4.1)and P n,< = the complement of P n,> . Our auxiliary state is defined as:1 N ∗ (cid:88) w ∈ P n,> t A ( w ) | w (cid:105) . (4.2)For t > Theorem 3.
The spectral gap of the t > , s > phase has an upper bound of s ) n t t − n / .Proof. We define a new state | ξ (cid:105) as: | ξ (cid:105) = 1 N ∗ (cid:88) w ∈ P n,> t A ( w ) P | w (cid:105) , (4.3)where N ∗ is the new normalization factor and the operator P sends the color c of the lastdown move of the longest prime walk to c + 1 mod s and leaves everything else unchanged.Because of the color imbalance we immediately have: (cid:104) ξ | GS (cid:105) = 0 , (4.4)6 FIG. 4: An representative walk in the superposition of | ξ (cid:105) that crosses the threshold of the cut-offwhen acted on by the operator ( t | ↓ rn − ↑ bn ↓ bn +1 (cid:105) − | ↑ bn − ↓ bn ↓ rn +1 (cid:105) )( (cid:104)↓ rn − ↑ bn ↓ bn +1 | t − (cid:104)↑ bn − ↓ bn ↓ rn +1 | ). and | ξ (cid:105) can be readily used as a variational wave function to bound the gap from above .Let us compute the variational energy associated with the | ξ (cid:105) state. First we note that: H ∂ | ξ (cid:105) = 0 , H X | ξ (cid:105) = 0 , (4.5)as each non-matching color pair is separated by at least n sites (while H X is only sensitiveto nearest neighbor violations). The same goes for most of the projectors in H F just theway it works in the ground state.However, in H F , we have also non-zero contributions coming from walks w that are one”Fredkin” move away from leaving the set P n , > . In other words, this happens when thefirst (second) projector in H F in Eq. (2.1) acts on the left (resp. right) endpoint of thelongest prime walk and changes its length from n + 1 to n − φ n − ,B (Eq. (2.7)) to the primewalk w corresponding to the one in Fig. 4 gives:11 + t (cid:104) w | ( t | ↓ rn − ↑ bn ↓ bn +1 (cid:105) − | ↑ bn − ↓ bn ↓ rn +1 (cid:105) )( (cid:104)↓ rn − ↑ bn ↓ bn +1 | t − (cid:104)↑ bn − ↓ bn ↓ rn +1 | ) | w (cid:105) = 11 + t , (4.6)and 11 + t (cid:104) w (cid:48) | ( t | ↓ rn − ↑ bn ↓ bn +1 (cid:105) − | ↑ bn − ↓ bn ↓ rn +1 (cid:105) )( (cid:104)↓ rn − ↑ bn ↓ bn +1 | t − (cid:104)↑ bn − ↓ bn ↓ rn +1 | ) | w (cid:105) = 0 , (4.7)7with w (cid:48) is any other walk in the | ξ (cid:105) (i.e. any other walk in P n,> ).We can now estimate the variational energy due to such paths. The number of thesepaths that will go from P n,> to P n,< when applying a Fredkin projector is very roughlybounded from above by 2 n s n (which is the total number of walks). On the other hand, theprobability amplitudes of a path that has a prime walk length of exactly n + 1 or n + 2 in P n,< , are penalized by their area differences from the highest weighted one, i.e. the shadedarea in Fig. 4, by a factor smaller than t − n / . We therefore have the following upper bound: (cid:104) ξ | H | ξ (cid:105) < s ) n t t − n / . (4.8)Thus we have proved an upper bound of exponential of square of system size on the spectralgap when t > , s > Remark:
The overall factor 2 above comes from possibility of modifying the prime pathon the left or on the right.
B. Exponential Upper Bound in the t > , s = 1 Phase
As has been discussed in the previous subsection, a bounded from above entanglemententropy is expected to be a strong indicator of the existence of a non-vanishing spectral gap.Yet that intuition fails in the t > , s = 1 phase of the Motzkin spin chain. The numericalresults in [20] showed the t > , s = 1 Motzkin chain is gapless despite the boundedness ofits entanglement entropy. Here we prove the Fredkin chain counterpart of this phenomenon,which can be readily adapted to the Motzkin chain.We follow the same strategy we used to construct low energy excitation state from the t > , s > Q − = { walks that starts from (0,0) and ends at (2n, -2) and never pass below x=-2. } Notice a Fredkin move acting on a walk in Q − always gives another walk in Q − . Theorem 4.
The spectral gap of the t > , s = 1 phase has an upper bound of t − n +1 . FIG. 5: Two representative walks in Q − . The light blue one can be shifted in the direction of theorange arrow to become the dark blue one with a relative weight increase of t n − corresponding tothe area of the shaded regions. Proof.
We define an excited state | η (cid:105) = 1 N (cid:63) (cid:88) w ∈ Q − t A ( w ) | w (cid:105) , (4.9)where N (cid:63) = ( (cid:80) w ∈ Q − t A ( w ) ) is the normalization factor. | η (cid:105) is clearly orthogonal to theground state as they have different total spins. Since | η (cid:105) only violates the boundary termin the Hamiltonian, after being acted on by H , only paths starting with a down movewill survive. To get an estimate on the amplitude of the paths left, we point out thatby rearranging the first down step to the last, (or equivalently shifting along the arrow inFig. 5,) we get another walk in Q − of area 2 n − (cid:104) η | H | η (cid:105) = (cid:80) w ∈ Q − t A ( w ) N (cid:63) < t − n +1 , (4.10)which gives an upper bound on the spectral gap. V. OUTLOOK
We mention a few other future directions worth exploring. While we have shown howto construct a multi-parameter deformation, we have only studied entropy and gap for a9uniform parameter t . This choice keeps the chain translationally invariant, however, nomomentum space arguments were involved in the analysis. A more general treatment willhave to contend with the distribution of the t A parameters.Second, the nature of the quantum phase transition is unclear. At a first glance, ithardly fits into the mechanism of symmetry breaking with an associated goldstone modeand exponents. To study the transition, as well as thermal effects, more detailed informationabout the density of states near the ground state is crucial. In particular, it would be veryinteresting to explore a possible field theoretic description in the continuous limit.For the colored case our variational upper bound on the gap gives an elementary wayof obtaining the gap behavior established in [19]. In [19] the colorful deformed Motzkinmodel was studied using more sophisticated mathematical machinery by utilizing the relationbetween frustration free local Hamiltonians and Markov chains, and applying a Cheegerinequality.Using a different variational wavefunction, we have also proven that the spectrum isgapless for the colorless version of the model at t >
1, in spite of entropy being bounded,furnishing an example of how bounded entanglement entropy does not imply a gap. Theidea can be applied immediately to the deformed Motzkin chain providing an explanationfor the surprising numerical observation of a vanishing gap in the t > , s = 1 phase [20].Finally, we remark that [20] also provides strong numerical evidence supporting the claimthat the spectrum will be gapped when 0 < t < s ). It would be interesting toestablish this observation rigorously. Acknowledgments:
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