We generalize the area-law violating models of Fredkin and Motzkin spin chains into two dimensions by building quantum six- and nineteen-vertex models with correlated interactions. The Hamiltonian is frustration free, and its projectors generate ergodic dynamics within the subspace of height configuration that are non negative. The ground state is a volume- and color-weighted superposition of classical bi-color vertex configurations with non-negative heights in the bulk and zero height on the boundary. The entanglement entropy between subsystems has a phase transition as the $q$ -deformation parameter is tuned, which is shown to be robust in the presence of an external field acting on the color degree of freedom. The ground state undergoes a quantum phase transition between area- and volume-law entanglement phases with a critical point where entanglement entropy scales as a function $L\log L$ of the linear system size $L$ . Intermediate power law scalings between $L\log L$ and $L^2$ can be achieved with an inhomogeneous deformation parameter that approaches 1 at different rates in the thermodynamic limit. For the $q>1$ phase, we construct a variational wave function that establishes an upper bound on the spectral gap that scales as $q^{-L^3/8}$ .We generalize the area-law violating models of Fredkin and Motzkin spin chains into two dimensions by building quantum six- and nineteen-vertex models with correlated interactions. The Hamiltonian is frustration free, and its projectors generate ergodic dynamics within the subspace of height configuration that are non negative. The ground state is a volume- and color-weighted superposition of classical bi-color vertex configurations with non-negative heights in the bulk and zero height on the boundary. The entanglement entropy between subsystems has a phase transition as the $q$-deformation parameter is tuned, which is shown to be robust in the presence of an external field acting on the color degree of freedom. The ground state undergoes a quantum phase transition between area- and volume-law entanglement phases with a critical point where entanglement entropy scales as a function $L\log L$ of the linear system size $L$. Intermediate power law scalings between $L\log L$ and $L^2$ can be achieved with an inhomogeneous deformation parameter that approaches 1 at different rates in the thermodynamic limit. For the $q>1$ phase, we construct a variational wave function that establishes an upper bound on the spectral gap that scales as $q^{-L^3/8}$.

arXiv:2210.03038Oct 6, 20221 citation

Quantum lozenge tiling and entanglement phase transition

Zhao Zhang, Israel Klich

While volume violation of area law has been exhibited in several quantum spin chains, the construction of a corresponding ground state in higher dimensions, entangled in more than one direction, has been an open problem. Here we construct a 2D frustration-free Hamiltonian with maximal violation of the area law. We do so by building a quantum model of random surfaces with color degree of freedom that can be viewed as a collection of colored Dyck paths. The Hamiltonian may be viewed as a 2D generalization of the Fredkin spin chain. It relates all the colored random surface configurations subject to a Dirichlet boundary condition and hard wall constraint from below to one another, and the ground state is therefore a superposition of all such classical states and non-degenerate. Its entanglement entropy between subsystems undergoes a quantum phase transition as the deformation parameter is tuned. The area- and volume-law phases are similar to the one-dimensional model, while the critical point scales with the linear size of the system L as Llog⁡L . Further it is conjectured that similar models with entanglement phase transitions can be built in higher dimensions with even softer area law violations at the critical point.While volume violation of area law has been exhibited in several quantum spin chains, the construction of a corresponding ground state in higher dimensions, entangled in more than one direction, has been an open problem. Here we construct a 2D frustration-free Hamiltonian with maximal violation of the area law. We do so by building a quantum model of random surfaces with color degree of freedom that can be viewed as a collection of colored Dyck paths. The Hamiltonian may be viewed as a 2D generalization of the Fredkin spin chain. It relates all the colored random surface configurations subject to a Dirichlet boundary condition and hard wall constraint from below to one another, and the ground state is therefore a superposition of all such classical states and non-degenerate. Its entanglement entropy between subsystems undergoes a quantum phase transition as the deformation parameter is tuned. The area- and volume-law phases are similar to the one-dimensional model, while the critical point scales with the linear size of the system L as Llog⁡L . Further it is conjectured that similar models with entanglement phase transitions can be built in higher dimensions with even softer area law violations at the critical point.

arXiv:2210.01098QuantumOct 10, 20249 citations

Exact rainbow tensor networks for the colorful Motzkin and Fredkin spin chains

Rafael N. Alexander, Amr A. Ahmadain, Zhao Zhang +1 more

We present bulk tensor networks that exactly represent the ground states of a continuous family of one-dimensional frustration-free Hamiltonians. These states, which are known as area-deformed Motzkin and Fredkin states, exhibit a novel quantum phase transition. By tuning a single parameter, they go from a phase obeying an area law to a highly entangled ``rainbow'' phase, where the half-chain entropy scales with the volume. Using the representation of these ground states as superpositions of random walks, we introduce tensor networks for these ground states where local and global rules of the walker are baked into bulk tensors, thereby providing an efficient description of the ground states (some of which satisfy a volume law scaling of entanglement entropy).We present bulk tensor networks that exactly represent the ground states of a continuous family of one-dimensional frustration-free Hamiltonians. These states, which are known as area-deformed Motzkin and Fredkin states, exhibit a novel quantum phase transition. By tuning a single parameter, they go from a phase obeying an area law to a highly entangled ``rainbow'' phase, where the half-chain entropy scales with the volume. Using the representation of these ground states as superpositions of random walks, we introduce tensor networks for these ground states where local and global rules of the walker are baked into bulk tensors, thereby providing an efficient description of the ground states (some of which satisfy a volume law scaling of entanglement entropy).

arXiv:1811.11974Physical review. B./Physical review. BDec 24, 201927 citations

Quantum phase transition from bounded to extensive entanglement entropy in a frustration-free spin chain

Zhao Zhang, Amr A. Ahmadain, Israel Klich

We introduce a continuous family of frustration-free Hamiltonians with exactly solvable ground states. We prove that the {ground state of our model is non-degenerate and exhibits} a novel quantum phase transition from bounded entanglement entropy to a massively entangled state with volume entropy scaling. The ground state may be interpreted as a deformation away from the uniform superposition of colored Motzkin paths, showed by Movassagh and Shor to have a large (square-root) but sub-extensive scaling of entanglement into a state with an extensive entropy.We introduce a continuous family of frustration-free Hamiltonians with exactly solvable ground states. We prove that the {ground state of our model is non-degenerate and exhibits} a novel quantum phase transition from bounded entanglement entropy to a massively entangled state with volume entropy scaling. The ground state may be interpreted as a deformation away from the uniform superposition of colored Motzkin paths, showed by Movassagh and Shor to have a large (square-root) but sub-extensive scaling of entanglement into a state with an extensive entropy.

arXiv:1606.07795Jun 24, 20166 citations

Exact critical exponents of the Motzkin and Fredkin chains

Olai B. Mykland, Zhao Zhang

The Motzkin and Fredkin chains are frustration-free spin models with exactly solvable ground states whose q q -deformations realize an exotic quantum phase transition from a disordered phase to an ordered one under domain-wall boundary conditions. Previous work has mainly focused on their entanglement scaling and spectral gaps, particularly in color-enriched variants. Here we systematically characterize the critical behavior of this transition with three main advances. First, we interpret bulk magnetic order as arising from correlations with boundary spins subject to an effective edge field, directly relating the ordered phase to the domain-wall boundary condition. Second, using the transfer matrix (TM) constructed from the exact matrix product state (MPS) representation, together with a continuum and RG analysis, we derive the algebraic decay of spin correlations and obtain the critical exponent \eta = \frac{3}{2} η = 3 2 . We further generalize the TM method to translationally invariant gapless states with generic power-law correlations, via a zero-dimensional Hamiltonian dual to the one-dimensional TM. Third, the TM spectrum reveals a duality between the ordered and disordered phases which, combined with scale invariance at criticality and the scaling dimension of the spin operator, yields \nu_± = \frac{2}{3} ν ± = 2 3 from an RG analysis of the q q -deformed ground states. Both exponents are confirmed numerically by MPS simulations and by direct diagonalization of the TM.The Motzkin and Fredkin chains are frustration-free spin models with exactly solvable ground states whose q q -deformations realize an exotic quantum phase transition from a disordered phase to an ordered one under domain-wall boundary conditions. Previous work has mainly focused on their entanglement scaling and spectral gaps, particularly in color-enriched variants. Here we systematically characterize the critical behavior of this transition with three main advances. First, we interpret bulk magnetic order as arising from correlations with boundary spins subject to an effective edge field, directly relating the ordered phase to the domain-wall boundary condition. Second, using the transfer matrix (TM) constructed from the exact matrix product state (MPS) representation, together with a continuum and RG analysis, we derive the algebraic decay of spin correlations and obtain the critical exponent \eta = \frac{3}{2} η = 3 2 . We further generalize the TM method to translationally invariant gapless states with generic power-law correlations, via a zero-dimensional Hamiltonian dual to the one-dimensional TM. Third, the TM spectrum reveals a duality between the ordered and disordered phases which, combined with scale invariance at criticality and the scaling dimension of the spin operator, yields \nu_± = \frac{2}{3} ν ± = 2 3 from an RG analysis of the q q -deformed ground states. Both exponents are confirmed numerically by MPS simulations and by direct diagonalization of the TM.

arXiv:2507.14656SciPost Physics CoreSep 1, 20261 citation

Deformed Fredkin spin chain with extensive entanglement

Olof Salberger, Takuma Udagawa, Zhao Zhang +3 more

Abstract We introduce a new spin chain which is a deformation of the Fredkin spin chain and has a phase transition between bounded and extensive entanglement entropy scaling. In this chain, spins have a local interaction of three nearest neighbors. The Hamiltonian is frustration-free and its ground state can be described analytically as a weighted superposition of Dyck paths that depends on a deformation parameter t . In the purely spin 1/2 case, whenever t ≠ 1 , the entanglement entropy obeys an area law: it is bounded from above by a constant, when the size of the block n increases. When a local color degree of freedom is introduced and t > 1 the entanglement entropy increases linearly, while for t < 1 an area law is obeyed. The half-chain entanglement entropy is tightly bounded by n log ⁡ s where 2 n is the length of the chain, and s is the number of colors. Our chain fosters a new example for a significant boost to entropy and for the existence of the associated critical rainbow phase where the entanglement entropy scales with volume that discovered by Zhang, Ahmadain and Klich ( Proc. Natl Acad. Sci 2017).Abstract We introduce a new spin chain which is a deformation of the Fredkin spin chain and has a phase transition between bounded and extensive entanglement entropy scaling. In this chain, spins have a local interaction of three nearest neighbors. The Hamiltonian is frustration-free and its ground state can be described analytically as a weighted superposition of Dyck paths that depends on a deformation parameter t . In the purely spin 1/2 case, whenever t ≠ 1 , the entanglement entropy obeys an area law: it is bounded from above by a constant, when the size of the block n increases. When a local color degree of freedom is introduced and t > 1 the entanglement entropy increases linearly, while for t < 1 an area law is obeyed. The half-chain entanglement entropy is tightly bounded by n log ⁡ s where 2 n is the length of the chain, and s is the number of colors. Our chain fosters a new example for a significant boost to entropy and for the existence of the associated critical rainbow phase where the entanglement entropy scales with volume that discovered by Zhang, Ahmadain and Klich ( Proc. Natl Acad. Sci 2017).

arXiv:1611.04983Journal of Statistical Mechanics Theory and ExperimentJun 1, 201759 citations

q

-deformation parameter is tuned, which is shown to be robust in the presence of an external field acting on the color degree of freedom. The ground state undergoes a quantum phase transition between area- and volume-law entanglement phases with a critical point where entanglement entropy scales as a function

L\log L

of the linear system size

L

. Intermediate power law scalings between

L\log L

and

L^2

can be achieved with an inhomogeneous deformation parameter that approaches 1 at different rates in the thermodynamic limit. For the

q>1

phase, we construct a variational wave function that establishes an upper bound on the spectral gap that scales as

q^{-L^3/8}