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Jainendra K Jain - One of the best experts on this subject based on the ideXlab platform.
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search for exact local hamiltonians for general fractional quantum hall states
Physical Review B, 2018Co-Authors: G J Sreejith, Mikael Fremling, Gun Sang Jeon, Jainendra K JainAbstract:We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. For this purpose, we study in detail the simplest non-trivial system beyond the Laughlin states, namely bosons at filling $\nu=\frac{2}{3}$ and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four particle interactions produces this state as the exact ground state, and speculate that this remains true even when interaction terms involving greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of $6\hbar$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $\lambda$-state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two sates have a high overlap; they have the same Root Partition; the quantum numbers for their neutral excitations are identical; and the quantum numbers for the quasiparticle excitations also match. On the quasihole side, we find that even though the quantum numbers of the lowest energy states agree with the prediction from the composite-fermion theory, these states are not separated from the others by a clearly identifiable gap. This prevents us from making a conclusive claim regarding the topological equivalence of the $\lambda$ state and the composite-fermion state. Our study illustrates how new candidate states can be identified from constraining selected many particle configurations and it would be interesting to pursue their topological classification.
G J Sreejith - One of the best experts on this subject based on the ideXlab platform.
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search for exact local hamiltonians for general fractional quantum hall states
Physical Review B, 2018Co-Authors: G J Sreejith, Mikael Fremling, Gun Sang Jeon, Jainendra K JainAbstract:We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. For this purpose, we study in detail the simplest non-trivial system beyond the Laughlin states, namely bosons at filling $\nu=\frac{2}{3}$ and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four particle interactions produces this state as the exact ground state, and speculate that this remains true even when interaction terms involving greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of $6\hbar$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $\lambda$-state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two sates have a high overlap; they have the same Root Partition; the quantum numbers for their neutral excitations are identical; and the quantum numbers for the quasiparticle excitations also match. On the quasihole side, we find that even though the quantum numbers of the lowest energy states agree with the prediction from the composite-fermion theory, these states are not separated from the others by a clearly identifiable gap. This prevents us from making a conclusive claim regarding the topological equivalence of the $\lambda$ state and the composite-fermion state. Our study illustrates how new candidate states can be identified from constraining selected many particle configurations and it would be interesting to pursue their topological classification.
Mikael Fremling - One of the best experts on this subject based on the ideXlab platform.
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search for exact local hamiltonians for general fractional quantum hall states
Physical Review B, 2018Co-Authors: G J Sreejith, Mikael Fremling, Gun Sang Jeon, Jainendra K JainAbstract:We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. For this purpose, we study in detail the simplest non-trivial system beyond the Laughlin states, namely bosons at filling $\nu=\frac{2}{3}$ and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four particle interactions produces this state as the exact ground state, and speculate that this remains true even when interaction terms involving greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of $6\hbar$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $\lambda$-state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two sates have a high overlap; they have the same Root Partition; the quantum numbers for their neutral excitations are identical; and the quantum numbers for the quasiparticle excitations also match. On the quasihole side, we find that even though the quantum numbers of the lowest energy states agree with the prediction from the composite-fermion theory, these states are not separated from the others by a clearly identifiable gap. This prevents us from making a conclusive claim regarding the topological equivalence of the $\lambda$ state and the composite-fermion state. Our study illustrates how new candidate states can be identified from constraining selected many particle configurations and it would be interesting to pursue their topological classification.
Gun Sang Jeon - One of the best experts on this subject based on the ideXlab platform.
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search for exact local hamiltonians for general fractional quantum hall states
Physical Review B, 2018Co-Authors: G J Sreejith, Mikael Fremling, Gun Sang Jeon, Jainendra K JainAbstract:We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. For this purpose, we study in detail the simplest non-trivial system beyond the Laughlin states, namely bosons at filling $\nu=\frac{2}{3}$ and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four particle interactions produces this state as the exact ground state, and speculate that this remains true even when interaction terms involving greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of $6\hbar$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $\lambda$-state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two sates have a high overlap; they have the same Root Partition; the quantum numbers for their neutral excitations are identical; and the quantum numbers for the quasiparticle excitations also match. On the quasihole side, we find that even though the quantum numbers of the lowest energy states agree with the prediction from the composite-fermion theory, these states are not separated from the others by a clearly identifiable gap. This prevents us from making a conclusive claim regarding the topological equivalence of the $\lambda$ state and the composite-fermion state. Our study illustrates how new candidate states can be identified from constraining selected many particle configurations and it would be interesting to pursue their topological classification.
Yonggao Chen - One of the best experts on this subject based on the ideXlab platform.
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on the r th Root Partition function ii
Journal of Number Theory, 2018Co-Authors: Yonggao ChenAbstract:Abstract Text For any positive real number r, let p r ( n ) be the number of solutions of the equation n = ⌊ a 1 r ⌋ + ⋯ + ⌊ a k r ⌋ with integers 1 ≤ a 1 ≤ ⋯ ≤ a k . Recently, Luca and Ralaivaosaona gave an asymptotic formula for p 2 ( n ) . In Part I, it is proved that, for any real number r > 1 , we have exp ( τ 1 n r / ( r + 1 ) ) ≤ p r ( n ) ≤ exp ( τ 2 n r / ( r + 1 ) ) for two positive constants τ 1 and τ 2 (depending only on r). In this paper, we prove that, for any real number r > 1 , p r ( n ) = exp ( c 1 n r / ( r + 1 ) + c 2 n ( r − 1 ) / ( r + 1 ) + ⋯ + c l n ( r − l + 1 ) / ( r + 1 ) + O ( n 1 / ( r + 1 ) ) ) , where l is the integer with l r ≤ l + 1 and c 1 , c 2 , … , c l are computable constants depending only on r. In particular, c 1 = ( 1 + r − 1 ) ( r ζ ( r + 1 ) Γ ( r + 1 ) ) 1 / ( r + 1 ) . For 0 r 1 , we pose a conjecture for the asymptotic formula for p r ( n ) . Video For a video summary of this paper, please visit https://youtu.be/Z9HiSbr9eJY .
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on the r th Root Partition function
Taiwanese Journal of Mathematics, 2016Co-Authors: Yonggao ChenAbstract:The well known Partition function $p(n)$ has a long research history, where $p(n)$ denotes the number of solutions of the equation $n = a_1 + \cdots + a_k$ with integers $1 \leq a_1 \leq \cdots \leq a_k$. In this paper, we investigate a new Partition function. For any real number $r > 1$, let $p_r(n)$ be the number of solutions of the equation $n = \lfloor \sqrt[r]{a_1} \rfloor + \cdots + \lfloor \sqrt[r]{a_k} \rfloor$ with integers $1 \leq a_1 \leq \cdots \leq a_k$, where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$. In this paper, it is proved that $\exp(c_1 n^{r/(r+1)}) \leq p_r(n) \leq \exp(c_2n^{r/(r+1)})$ for two positive constants $c_1$ and $c_2$ (depending only $r$).
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on the square Root Partition function
Comptes Rendus Mathematique, 2015Co-Authors: Yonggao ChenAbstract:Abstract The well-known Partition function p ( n ) , which is the number of solutions of the equation n = a 1 + ⋯ + a k with integers 1 ≤ a 1 ≤ ⋯ ≤ a k , has a long research history. In this note, we investigate a new Partition function. Let q ( n ) be the number of solutions of the equation n = [ a 1 ] + ⋯ + [ a k ] with integers 1 ≤ a 1 ≤ ⋯ ≤ a k , where [ x ] denotes the integral part of x. We prove that exp ( c 1 n 2 / 3 ) ≤ q ( n ) ≤ exp ( c 2 n 2 / 3 ) for two explicit positive constants c 1 and c 2 .
