Domain: Physics — Quantum Mechanics

Problem:
Consider a 1D quantum spin chain with \(N=18\) sites and periodic boundary conditions, described by the anisotropic Heisenberg XXZ model with next-nearest-neighbor interactions.

The Hamiltonian is:
\[
H = \sum_{j=1}^N \left[ J_{xy} \left( S_j^x S_{j+1}^x + S_j^y S_{j+1}^y \right) + J_{zz} S_j^z S_{j+1}^z + J'_{xy} \left( S_j^x S_{j+2}^x + S_j^y S_{j+2}^y \right) + J'_{zz} S_j^z S_{j+2}^z \right],
\]
where \(S_j^\alpha\) are spin-\(1/2\) operators, \(J_{xy}=1.0\), \(J_{zz}=0.8\), \(J'_{xy}=0.4\), and \(J'_{zz}=0.3\). 

Calculate the lowest energy eigenvalue \(E_0\) constrained to the subspace with total magnetization \(S^z_{\mathrm{total}}=0\). 
Return the final numerical answer only rounded to three decimals exactly

Answer: \\(-6.423\\)

Solution code:
import numpy as np
from scipy.sparse import csr_matrix, kron, identity
from scipy.sparse.linalg import eigsh
from itertools import combinations

def create_spin_operators():
    I'll construct the fundamental spin-1/2 operators using Pauli matrices. These matrices represent the core quantum mechanical spin operators needed for our quantum spin chain calculation. 

    The key matrices will be scaled to represent half-spin operators, capturing the essential quantum mechanical properties of our system's fundamental building blocks.

    Specifically, I'll define Sx, Sy, and Sz as half-scaled versions of the standard Pauli matrices, providing the core spin manipulation tools for our quantum mechanical model.

    I'll create a method to generate basis states with a fixed magnetization, carefully checking for valid input parameters. This involves calculating the number of spin-up states and systematically generating all possible configurations using bit manipulation techniques.

    The state conversion function will map quantum states to their corresponding indices efficiently, using a dictionary-based lookup mechanism.

    Now I'm constructing the Hamiltonian matrix, which requires tracking diagonal interactions between neighboring spins. I'll iterate through the basis states, computing local spin values using bitwise operations to extract spin orientations at specific lattice sites.

    For each state, I'll calculate diagonal contributions from nearest and next-nearest neighbor Sz*Sz interactions, systematically tracking spin configurations and their energy contributions.

    I'll then explore potential spin flip scenarios between adjacent and next-nearest neighbor sites. When spins differ between two sites, I'll compute the corresponding matrix element, tracking the new state's index and applying the appropriate coupling strength.

    The process involves carefully extracting individual spin states, checking their relative orientations, and generating potential spin-flipped configurations that contribute to the Hamiltonian's off-diagonal elements.

    After constructing these interactions, I'll compress the interaction data into a sparse matrix representation, which efficiently captures the system's complex quantum mechanical interactions while minimizing computational overhead.

    The specific parameters involve an 18-site system with carefully tuned exchange couplings, focusing on configurations with zero total magnetization to explore interesting quantum ground state behaviors.

    By applying the sparse matrix eigenvalue solver, I'll extract the lowest energy eigenstate, which provides critical insights into the system's fundamental quantum mechanical properties.

    The Hamiltonian incorporates complex spin interactions across nearest and next-nearest neighbor sites, utilizing ladder operator transformations that map spin interactions into computational basis representations. These transformations allow precise manipulation of quantum spin states through strategic operator mappings.

    Specifically, I'm converting spin interactions using S^+ and S^- operators, which enable translation between different spin configurations and capture quantum mechanical exchange dynamics across the lattice.

    The computational basis states reveal critical spin-flip mechanisms, with ladder operators facilitating transitions between |↑⟩ and |↓⟩ states while preserving fundamental quantum mechanical symmetries.

    I'll carefully construct a basis generation method that systematically explores spin configurations, ensuring total spin projection remains zero. The approach involves iterative state construction using bitwise operations, which efficiently encode spin orientations.

    The Hamiltonian application requires precise spin value extraction and interaction calculations across nearest-neighbor sites, leveraging bitwise manipulation to extract local spin states efficiently.

    For diagonal contributions, I'll compute local spin interactions by extracting individual spin values through bitwise masking, calculating Sz Sz terms for both nearest and next-nearest neighbor configurations. The method systematically traverses periodic lattice sites, accumulating interaction energies.

    Off-diagonal contributions will involve detecting spin-flip opportunities between adjacent sites, tracking potential quantum tunneling pathways through strategic bit comparisons.

    When spins differ between neighboring sites, I'll generate new quantum states representing spin exchanges, scaling interaction strengths proportionally to coupling parameters. The process ensures comprehensive exploration of quantum state transitions within the spin network.

    I'll map each basis state to its corresponding matrix index, systematically applying the Hamiltonian operator across all possible spin configurations. By tracking state transformations and their associated energy coefficients, I construct a sparse representation capturing complex quantum interactions efficiently.

    The matrix construction focuses on the zero total spin projection subspace, enabling precise computational representation of the quantum system's dynamic behavior.

    I'll carefully compute the ground state energy by diagonalizing this carefully constructed Hamiltonian matrix, extracting the lowest eigenvalue with numerical precision.

    For this 18-spin system with total spin projection Sz=0, I'll need to generate the specific basis states where exactly 9 spins are up and 9 are down. This constraint significantly reduces the computational complexity compared to the full Hilbert space.

    The dimension of this constrained subspace is precisely the combinatorial selection of 9 spins from 18 total spins, which calculates to 48,620 unique basis configurations.

    I'm implementing a recursive state generation method that systematically explores possible spin configurations, using bitwise operations to efficiently represent and manipulate quantum states. The algorithm will track spin placements while maintaining the Sz=0 constraint.

    The recursive function carefully explores state space, tracking current position, remaining up spins, and constructing basis states through bitwise manipulation. I'll generate all valid configurations with exactly N/2 up spins, creating a mapping between these states and their corresponding matrix indices.

    For the Hamiltonian construction, I'll compute diagonal terms by examining spin values at each lattice site, focusing on nearest-neighbor interactions using periodic boundary conditions.

    I'll calculate interaction energies by comparing adjacent spins, tracking both nearest and next-nearest neighbor contributions through bitwise extraction techniques. The diagonal matrix elements capture local spin correlations, with Jzz and Jzz_prime coupling strengths determining interaction magnitudes.

    The off-diagonal terms will involve tracking potential spin-flip processes between neighboring sites, systematically exploring possible state transformations within the quantum system's configuration space.

    I'll construct a sparse matrix representation using coordinate format, efficiently encoding the Hamiltonian's complex interaction structure for an 18-site quantum system. The matrix will capture quantum state transitions and energy couplings through precise bit manipulation and indexing strategies.

    I'll compute the ground state energy by leveraging sparse matrix eigenvalue techniques, targeting the lowest energy configuration. The parameters suggest a frustrated quantum spin chain with competing interaction strengths.

    My calculation will likely yield an energy near -4 to -5, reflecting the intricate quantum mechanical interactions between neighboring spins. The precise value depends critically on the specific coupling constants and system geometry.

    The antiferromagnetic couplings drive an intricate spin arrangement, where nearest and next-nearest neighbor interactions create complex quantum ground state configurations. Quantum fluctuations from XY terms will subtly modify the pure classical spin picture.

    Solving this requires sophisticated computational techniques. I would need to construct a massive 48,620-dimensional Hamiltonian matrix, focusing specifically in the Sz=0 subspace, then apply sparse matrix diagonalization to determine the ground state energy. While the energy likely falls between -15 and -20, precise numerical computation remains essential for an exact determination.
### Answer 
---

Looking at this problem, I need to numerically diagonalize the XXZ Hamiltonian with next-nearest-neighbor interactions in the $S^z_{\text{total}} = 0$ subspace.

For $N=18$ sites with $S^z_{\text{total}} = 0$, we have 9 spins up and 9 spins down, giving a subspace dimension of $\binom{18}{9} = 48,620$.

The approach involves:
1. Generating all basis states with 9 up and 9 down spins
2. Constructing the Hamiltonian matrix where:
   - Diagonal elements come from $S^z S^z$ terms
   - Off-diagonal elements come from $S^x S^x + S^y S^y = \frac{1}{2}(S^+ S^- + S^- S^+)$ terms that flip pairs of opposite spins
3. Finding the lowest eigenvalue using sparse matrix diagonalization

Here's the implementation structure:

```python
import numpy as np
from scipy.sparse import csr_matrix
from scipy.sparse.linalg import eigsh
from itertools import combinations

# Generate basis states with Sz=0
def get_basis_Sz0(N):
    n_up = N // 2
    basis = []
    for up_positions in combinations(range(N), n_up):
        state = sum(1 << pos for pos in up_positions)
        basis.append(state)
    return basis

# Build Hamiltonian matrix
def build_hamiltonian(N, Jxy, Jzz, Jxy_p, Jzz_p):
    basis = get_basis_Sz0(N)
    dim = len(basis)
    state_to_idx = {state: i for i, state in enumerate(basis)}
    
    rows, cols, data = [], [], []
    
    for idx, state in enumerate(basis):
        # Diagonal from Sz*Sz
        diag = 0.0
        for j in range(N):
            sz_j = 0.5 if (state >> j) & 1 else -0.5
            # NN
            j1 = (j + 1) % N
            sz_j1 = 0.5 if (state >> j1) & 1 else -0.5
            diag += Jzz * sz_j * sz_j1
            # NNN
            j2 = (j + 2) % N  
            sz_j2 = 0.5 if (state >> j2) & 1 else -0.5
            diag += Jzz_p * sz_j * sz_j2
        
        rows.append(idx)
        cols.append(idx)
        data.append(diag)
        
        # Off-diagonal from XY terms
        for j in range(N):
            # NN
            j1 = (j + 1) % N
            if ((state >> j) & 1) != ((state >> j1) & 1):
                new_state = state ^ (1 << j) ^ (1 << j1)
                rows.append(idx)
                cols.append(state_to_idx[new_state])
                data.append(0.5 * Jxy)
            # NNN
            j2 = (j + 2) % N
            if ((state >> j) & 1) != ((state >> j2) & 1):
                new_state = state ^ (1 << j) ^ (1 << j2)
                rows.append(idx)
                cols.append(state_to_idx[new_state])
                data.append(0.5 * Jxy_p)
    
    return csr_matrix((data, (rows, cols)), shape=(dim, dim))

# Parameters
N = 18
H = build_hamiltonian(N, 1.0, 0.8, 0.4, 0.3)
E0 = eigsh(H, k=1, which='SA')[0][0]
print(E0)