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arXiv:2202.00564

Machine learning approach to study quantum phases of a frustrated one dimensional spin-1/2 system

Sk Saniur Rahaman, Sumit Haldar, Manoranjan Kumar

Published Feb 1, 2022• 0 citations
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Abstract

Frustration driven quantum fluctuation leads to many exotic phases in the ground state and study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. Here, a frustrated Heisenberg $J_1-J_2$ model of spin-1/2 chain with nearest exchange interaction $J_1$ and next nearest exchange interaction $J_2$ is studied using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSC) of ground-state (GS) and first excited state (FES) for different $J_2/J_1$ are used as the input in PCA to construct the co-variance matrix. The `quantified principal component' of the largest eigenvalue of co-variance matrix $p_1(J_2/J_1)$ is calculated and it is shown that the nature and variation of $p_1(J_2/J_1)$ can accurately predict the phase transitions and degeneracies in the GS. The $p_1(J_2/J_1)$ calculated from the MPSC of GS can only exhibit the signature of degeneracies in the GS, whereas, $p_1(J_2/J_1)$ calculated from MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that jump in $p_1(J_2/J_1)$ of FES at $J_2/J_1 \approx 0.241$, indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of first two principal components of FES shows distinct band formation for different phases.

DOI
10.48550/arxiv.2202.00564

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Machine learning approach to study quantum phase transitions of a frustrated one dimensional spin-1/2 system

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Abstract Frustration-driven quantum fluctuation leads to many exotic phases in the ground state (GS) and the study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. We study a frustrated Heisenberg J 1 − J 2 model of spin-1/2 chain with nearest exchange interaction J 1 and next nearest exchange interaction J 2 using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSCs) of GS and first excited state (FES) for different J 2 / J 1 are used as the input in PCA to construct the covariance matrix. The ‘quantified principal component’ p 1 ( J 2 / J 1 ) of the largest eigenvalue of the covariance matrix is calculated and it is shown that the nature and variation of p 1 ( J 2 / J 1 ) can accurately predict the phase transitions and degeneracies in the GS. The p 1 ( J 2 / J 1 ) calculated from the MPSC of GS only exhibits the signature of degeneracies in the GS, whereas, p 1 ( J 2 / J 1 ) calculated from the MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that the jump in p 1 ( J 2 / J 1 ) of FES at J 2 / J 1 ≈ 0.241 , indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of the first two principal components of FES shows distinct band formation for different phases. The MPSCs are obtained by using an iterative variational method (IVM) which gives the approximate eigenvalues and eigenvectors.

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Frustration driven quantum fluctuation leads to many exotic phases in the ground state and study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. Here, a frustrated Heisenberg J1−J2J_1-J_2J1​−J2​ model of spin-1/2 chain with nearest exchange interaction J1J_1J1​ and next nearest exchange interaction J2J_2J2​ is studied using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSC) of ground-state (GS) and first excited state (FES) for different J2/J1J_2/J_1J2​/J1​ are used as the input in PCA to construct the co-variance matrix. The `quantified principal component' of the largest eigenvalue of co-variance matrix p1(J2/J1)p_1(J_2/J_1)p1​(J2​/J1​) is calculated and it is shown that the nature and variation of p1(J2/J1)p_1(J_2/J_1)p1​(J2​/J1​) can accurately predict the phase transitions and degeneracies in the GS. The p1(J2/J1)p_1(J_2/J_1)p1​(J2​/J1​) calculated from the MPSC of GS can only exhibit the signature of degeneracies in the GS, whereas, p1(J2/J1)p_1(J_2/J_1)p1​(J2​/J1​) calculated from MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that jump in p1(J2/J1)p_1(J_2/J_1)p1​(J2​/J1​) of FES at J2/J1≈0.241J_2/J_1 \approx 0.241J2​/J1​≈0.241, indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of first two principal components of FES shows distinct band formation for different phases.
Abstract Frustration-driven quantum fluctuation leads to many exotic phases in the ground state (GS) and the study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. We study a frustrated Heisenberg J 1 − J 2 model of spin-1/2 chain with nearest exchange interaction J 1 and next nearest exchange interaction J 2 using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSCs) of GS and first excited state (FES) for different J 2 / J 1 are used as the input in PCA to construct the covariance matrix. The ‘quantified principal component’ p 1 ( J 2 / J 1 ) of the largest eigenvalue of the covariance matrix is calculated and it is shown that the nature and variation of p 1 ( J 2 / J 1 ) can accurately predict the phase transitions and degeneracies in the GS. The p 1 ( J 2 / J 1 ) calculated from the MPSC of GS only exhibits the signature of degeneracies in the GS, whereas, p 1 ( J 2 / J 1 ) calculated from the MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that the jump in p 1 ( J 2 / J 1 ) of FES at J 2 / J 1 ≈ 0.241 , indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of the first two principal components of FES shows distinct band formation for different phases. The MPSCs are obtained by using an iterative variational method (IVM) which gives the approximate eigenvalues and eigenvectors.
We apply unsupervised machine learning techniques, mainly principal component analysis (PCA), to compare and contrast the phase behavior and phase transitions in several classical spin models-the square- and triangular-lattice Ising models, the Blume-Capel model, a highly degenerate biquadratic-exchange spin-1 Ising (BSI) model, and the two-dimensional XY model-and we examine critically what machine learning is teaching us. We find that quantified principal components from PCA not only allow the exploration of different phases and symmetry-breaking, but they can distinguish phase-transition types and locate critical points. We show that the corresponding weight vectors have a clear physical interpretation, which is particularly interesting in the frustrated models such as the triangular antiferromagnet, where they can point to incipient orders. Unlike the other well-studied models, the properties of the BSI model are less well known. Using both PCA and conventional Monte Carlo analysis, we demonstrate that the BSI model shows an absence of phase transition and macroscopic ground-state degeneracy. The failure to capture the "charge" correlations (vorticity) in the BSI model (XY model) from raw spin configurations points to some of the limitations of PCA. Finally, we employ a nonlinear unsupervised machine learning procedure, the "autoencoder method," and we demonstrate that it too can be trained to capture phase transitions and critical points.
Studying quantum phase transitions through order parameters is a traditional method, but studying phase transitions by machine learning is a brand new field. The ability of machine learning to classify, identify, or interpret massive data sets may provide physicists with similar analyses of the exponentially large data sets embodied in the Hilbert space of quantum many-body system. In this work, we propose a method of using unsupervised learning algorithm of the Gaussian mixture model to classify the state vectors of the J1-J2 antiferromagnetic Heisenberg spin chain system, then the supervised learning algorithm of the convolutional neural network is used to identify the classification point given by the unsupervised learning algorithm, and the cross-validation method is adopted to verify the learning effect. Using this method, we study the J1-J2 Heisenberg spin chain system with chain length N = 8, 10, 12, 16 and obtain the same conclusion. The first order phase transition point of J1-J2 antiferromagnetic Heisenberg spin chain system can be accurately found from the ground state vector, but the infinite order phase transition point cannot be found from the ground state vector. The first order and the infinite order phase transition point can be found from the first excited state vector, which indirectly shows that the first excited state may contain more information than the ground state of J1-J2 antiferromagnetic Heisenberg spin chain system. The visualization of the state vector shows the reliability of the machine learning algorithm, which can extract the feature information from the state vector. The result reveals that the machine learning techniques can directly find some possible phase transition points from a large set of state vectorwithout prior knowledge of the energy or locality conditions of the Hamiltonian, which may assists us in studying unknown systems. Supervised learning can verify the phase transition points given by unsupervised learning, thereby indicating that we can discover some useful information about unknown systems only through machine learning techniques. Machine learning techniques can be a basic research tool in strong quantum-correlated systems, and it can be adapted to more complex systems, which can help us dig up hidden information.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
The spin-1/2 chain with isotropic Heisenberg exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1. Its quantum phase diagram has critical points at fixed J1/J2 between gapless phases with nondegenerate ground state (GS) and quasi-long-range order (QLRO) and gapped phases with doubly degenerate GS and spin correlation functions of finite range. In finite chains, exact diagonalization (ED) estimates critical points as level crossing of excited states. GS spin correlations enter in the spin structure factor S(q) that diverges at wave vector qm in QLRO(q(m)) phases with periodicity 2π/q(m) but remains finite in gapped phases. S(q(m)) is evaluated using ED and density matrix renormalization group (DMRG) calculations. Level crossing and the magnitude of S(q(m)) are independent and complementary probes of quantum phases, based respectively on excited and ground states. Both indicate a gapless QLRO(π/2) phase between -1.2 < J1/|J2| < 0.45. Numerical results and field theory agree well for quantum critical points at small frustration J2 but disagree in the sector of weak exchange J1 between Heisenberg antiferromagnetic chains on sublattices of odd and even-numbered sites.
The quantum phases of one-dimensional spin s=1/2s=1/2s=1/2 chains are discussed for models with two parameters, frustrating exchange g=J2>0g={J}_{2}>0g=J2​>0 between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at g=0g=0g=0 has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin S=SA+SBS={S}_{A}+{S}_{B}S=SA​+SB​, have singlet GS for any ggg, $\ensuremath{\alpha}\ensuremath{\ge}0$ and decouple at 1/g=01/g=01/g=0 to linear Heisenberg antiferromagnets on sublattices AAA and BBB of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap Em{E}_{m}Em​ to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at gc=1/4ln2{g}_{c}=1/4\mathrm{ln}2gc​=1/4ln2 from a GS with perfect LRO to a decoupled phase with SA=SB=0{S}_{A}={S}_{B}=0SA​=SB​=0 for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.