Strong Disorder Renormalization Group Method for Bond Disordered Antiferromagnetic Quantum Spin Chains with Long Range Interactions: Excited States and Finite Temperature Properties

This paper extends the strong disorder renormalization group method to analyze excited states and finite temperature properties of bond-disordered antiferromagnetic quantum spin chains with both short-range and long-range power-law interactions, deriving key thermodynamic and entanglement characteristics while characterizing the distribution of coupling signs and amplitudes.

The Gist

Imagine a long line of tiny, spinning tops (quantum spins) scattered randomly along a wire. Some are spinning up, some down. In a perfect world, they would all pair up neatly with their immediate neighbors. But in this messy, "disordered" world, they are jostling, and the strength of their connection depends on how far apart they are.

This paper is about a new way to understand how these chaotic spin chains behave, especially when they are hot (finite temperature) and when they can "talk" to each other over long distances, not just to their neighbors.

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Problem: A Chaotic Dance Floor

Think of the spin chain as a crowded dance floor.

• The Spins: The dancers.
• The Bonds: How strongly two dancers want to hold hands.
• The Disorder: The dancers are placed randomly. Some are very close; some are far apart.
• The Interaction: In this specific model, the "dance" is an antiferromagnetic one. This means if two dancers hold hands, they prefer to face opposite directions (one up, one down) to be happy.

The challenge is that the "music" (temperature) is playing, and the dancers are jittery. We want to know: Who ends up holding hands? What is the energy of the crowd? And how "entangled" (connected) are they?

2. The Tool: The "Strong Disorder" Renormalization Group (SDRG)

To solve this, the author uses a method called SDRG. Imagine this as a game of "Musical Chairs" for pairs.

1. Find the loudest music: You look at the whole line and find the two dancers who are holding hands the tightest (the strongest bond).
2. Freeze them: You decide, "Okay, these two are a team. They are done dancing with anyone else for now." You lock them into a specific state (either a perfect pair or a slightly messy trio).
3. The Ripple Effect: When you lock these two in, it changes how the next two dancers (who were waiting for them) feel about each other. It's like removing a pillar from a building; the walls next to it have to adjust.
4. Repeat: You do this over and over, pairing up the strongest bonds, then the next strongest, until everyone is paired up.

This method is great for the "ground state" (the coldest, calmest state). But the author asked: "What happens when the dance floor is hot?"

3. The New Discovery: Hot Dance Floors and Long-Distance Calls

The author extended this method to study excited states (hotter, more energetic states) and long-range interactions (where a dancer can reach across the room to hold hands with someone far away, not just the person next to them).

The Short-Range Case (Neighbors Only)

First, they looked at a chain where dancers only hold hands with their immediate neighbors.

• The Result: Even when it's hot, the "tightest pairs" still follow the same rules as the cold ones. The distribution of how strong the bonds are remains the same.
• The Twist: However, the direction of the hold changes. In the cold state, everyone holds hands "anti-parallel" (opposite directions). In the hot state, some pairs start holding hands "parallel" (same direction) just by chance. As the temperature rises, more and more of these "wrong-way" pairs appear.

The Long-Range Case (Reaching Across the Room)

This is the big innovation. Here, a dancer at one end can hold hands with someone at the other end. The strength of the hold drops off with distance (like a power law).

• The Math: The author derived a complex "Master Equation" (a rulebook) for how these long-distance bonds change as you pair up the strongest ones.
• The Finding:
  • If the interaction drops off fast (large power exponent, $\alpha > 2$), the system behaves nicely. The "hot" state looks very similar to the "cold" state. The distribution of bond strengths is predictable.
  • If the interaction drops off slowly (small $\alpha < 2$), things get weird. The "ripple effect" of pairing up one couple creates new, strange types of interactions (like three-way connections) that the simple pairing method can't handle. The system becomes a "Rainbow State," where pairs overlap in complex, arching shapes rather than simple lines.

4. What Does This Mean for Real Physics?

The author calculated what we can actually measure in a lab:

• Magnetic Susceptibility (How much they react to a magnet):

  • In these hot, disordered chains, the magnetic response is dominated by "free" spins that aren't perfectly paired.
  • It follows a Curie Law (a standard rule for magnets), but the strength of this effect depends on how "long-range" the interactions are. If the interactions are very long-range, the magnetic response changes in a specific, predictable way.

• Entanglement Entropy (How connected the system is):

  • Entanglement is like a secret handshake between particles that links their fates.
  • In the cold ground state, the "secret handshake" grows logarithmically (slowly but steadily) as you look at bigger chunks of the chain.
  • The Hot Surprise: When the temperature gets very high, this connection gets cut in half. The "secret handshake" becomes less effective because the heat is shaking the dancers too much to maintain a deep connection.

5. The Big Picture Analogy

Imagine a city where people are connected by telephone lines of varying lengths.

• The SDRG method is like a phone company that keeps cutting the strongest lines first to see how the network reorganizes.
• The Paper's Insight:
  • If the city is cold (low temp), the network reorganizes in a very orderly, predictable way.
  • If the city is hot (high temp), the network still reorganizes similarly, but some people start calling the "wrong" number (sign changes).
  • If the phone lines are very long (long-range), the network is robust and predictable unless the lines are too long and weak. If they are too long, the network gets messy, and you need a new rulebook to understand how the calls are routed.

Summary

This paper takes a powerful mathematical tool (SDRG) and upgrades it to handle heat and long-distance connections in quantum magnets.

• Good news: For most "normal" long-range interactions, the system is stable and predictable even when hot.
• Bad news: If the interactions are very long-range and weak, the system gets chaotic, requiring new physics to understand.
• Key takeaway: Heat doesn't destroy the "random singlet" structure of these magnets, but it does weaken the deep quantum connections (entanglement) between them.

Technical Summary

Here is a detailed technical summary of the paper "Strong Disorder Renormalization Group Method for Bond Disordered Antiferromagnetic Quantum Spin Chains with Long Range Interactions: Excited States and Finite Temperature Properties" by S. Kettemann.

1. Problem Statement

The paper addresses the challenge of deriving thermodynamic and dynamic properties for bond-disordered antiferromagnetic quantum spin chains with long-range power-law interactions ($J_{ij} \propto r_{ij}^{-\alpha}$) at finite temperatures.

While the Strong Disorder Renormalization Group (SDRG) method is well-established for studying the ground states of short-range disordered systems (yielding the Infinite Randomness Fixed Point, IRFP) and has recently been extended to long-range ground states, a rigorous treatment of excited states and finite temperature ($T > 0$) properties for long-range coupled systems was lacking. Specifically, the authors aim to understand how thermal fluctuations affect the distribution of coupling signs, the nature of excited states, and physical observables like magnetic susceptibility and entanglement entropy in these systems.

2. Methodology

The author extends the SDRG-X method (a real-space representation of SDRG designed for excited states) to include finite temperature and long-range interactions.

• Model: A chain of $N$ spin-1/2 particles with random positions and antiferromagnetic XX-interactions decaying as $J_{ij} = J_0 |(r_i - r_j)/a|^{-\alpha}$.
• SDRG-X Procedure:
  • The system is iteratively decimated by identifying the strongest coupled pair $(i, j)$ with energy scale $\Omega$.
  • Unlike ground-state SDRG (which assumes the pair is always in the singlet state), SDRG-X considers all four possible pair states: the singlet ($|0\rangle$) and three triplet states ($|1\rangle, |2\rangle, |3\rangle$).
  • Thermal Occupation: At a given RG scale $\Omega$ and temperature $T$, the probability $p_s$ of the pair occupying a specific state $s$ is determined by the Boltzmann distribution: $p_s \propto e^{-\beta E_s}$.
  • Renormalization Rules: The decimation of a pair generates new effective couplings between neighboring spins ($l, m$). Crucially, the sign of the new coupling depends on the state $s$ of the decimated pair:
    • Singlet ($s=0$) and Entangled Triplet ($s=3$): Preserve the antiferromagnetic sign ($\sigma = +$).
    • Unentangled Triplets ($s=1, 2$): Flip the sign to ferromagnetic ($\sigma = -$).
• Master Equations: The authors derive Master equations for the distribution function of coupling distances $P_{\sigma, T}(\tilde{r}, \Omega)$, tracking both the magnitude and the sign ($\sigma = \pm$) of the couplings as the energy scale $\Omega$ is lowered.

3. Key Contributions and Results

A. Short-Range Coupling Limit ($\alpha \to \infty$)

• Coupling Distribution: The distribution of coupling amplitudes follows the standard Infinite Randomness Fixed Point (IRFP) distribution, which is independent of temperature.
• Sign Distribution: While the ground state is purely antiferromagnetic, finite temperature introduces ferromagnetic couplings. As $T$ increases, the probability of negative couplings increases.
  • At high temperatures ($T \gg \Omega$), the sign becomes random ($P_+ = P_- = 1/2$).
  • At low temperatures ($T \ll \Omega$), the system remains predominantly antiferromagnetic.
• Physical Implication: For short-range models, the sign of the coupling does not affect the energy spectrum (due to symmetry in the XX model), so the IRFP distribution of amplitudes remains valid for excited states.

B. Long-Range Coupling ($J_{ij} \propto r^{-\alpha}$)

• Regime $\alpha \gg 2$:
  • The distribution of coupling amplitudes retains the strong disorder form but with a finite width $\Gamma = 2\alpha$.
  • The sign distribution is derived, showing that for $\alpha > 2$, the probability of sign flipping is small, and the system remains dominated by antiferromagnetic correlations.
  • The SDRG condition (renormalized distance $\tilde{r} > \rho$) holds, validating the pair-based decimation scheme.
• Regime $\alpha < 2$:
  • Breakdown of Pair Approximation: The renormalized coupling can exceed the RG scale $\Omega$ for certain distances, violating the standard SDRG condition.
  • New Terms: The renormalization generates 3-point interactions and couplings between spin $z$-components and pseudo-spins acting on the degenerate triplet subspace.
  • Rainbow States: Numerical evidence suggests that for $\alpha < 2$, excited states are no longer random singlet pairs but form "rainbow states" (overlapping pairs), requiring an extension of the SDRG method to multi-spin clusters.

C. Finite Temperature Physical Properties

Using the derived distribution functions, the author calculates key observables:

1. Magnetic Susceptibility ($\chi$):

   • The susceptibility is dominated by paramagnetic $S=1$ spins (unentangled triplets) rather than free $S=1/2$ spins.
   • Short-range: $\chi(T) \sim \frac{1}{T \ln^2(\Omega_0/T)}$.
   • Long-range ($\alpha > 1$): $\chi(T) \sim T^{\frac{1}{\alpha} - 1} + \frac{c}{T}$. The Curie term ($1/T$) from$S=1$pairs dominates, with a weight that decreases as$\alpha$ increases.

2. Singlet Length Distribution:

   • The distribution of pair lengths $P(l)$ follows a power law $P(l) \sim l^{-2}$, identical to the ground state result, and remains valid at finite temperature for $\alpha \gg 1$.

3. Spin Correlation and Concurrence:

   • The ensemble-averaged concurrence decays as $\langle C_n \rangle \sim n^{-2}$.
   • The typical correlation function decays as $C_{typ}(n) \sim n^{-(2+\alpha)}$ for long-range models. These results hold at finite $T$ because the pair length distribution is unchanged.

4. Entanglement Entropy ($S_n$):

   • Ground State: $S_n \sim \frac{1}{6} \ln 2 \ln n$ (effective central charge $\bar{c} = \ln 2$).
   • Finite Temperature: The entanglement entropy still grows logarithmically with subsystem size $n$, but with a temperature-dependent correction.
   • High Temperature Limit ($T \to \infty$): The effective central charge is halved to $\bar{c}_{T \to \infty} = \frac{1}{2} \ln 2$. This indicates a reduction in entanglement due to thermal mixing of states.

5. Entanglement Growth after Quantum Quench:

   • For a global quench from a Néel state, the entanglement entropy grows as $S(t) \sim \frac{1}{2\alpha} \ln t$. This is faster than the double-logarithmic growth ($\ln \ln t$) found in short-range random chains.

4. Significance

• Extension of SDRG: The paper successfully generalizes the SDRG-X method to finite temperatures and long-range interactions, providing a non-perturbative analytical framework for disordered quantum systems where exact diagonalization fails.
• Thermal Effects on Disorder: It reveals that while the amplitude distribution of couplings in long-range systems is robust (finite width fixed point), the sign distribution is highly sensitive to temperature, leading to a proliferation of ferromagnetic bonds at high $T$.
• Phase Transition in Methodology: It identifies a critical threshold at $\alpha = 2$. For $\alpha < 2$, the standard pair-based SDRG breaks down due to the emergence of multi-spin interactions and rainbow states, highlighting a fundamental change in the nature of the excited states.
• Experimental Relevance: The results are directly applicable to emerging quantum simulators (trapped ions, Rydberg atoms, NV centers) where long-range interactions and tunable disorder can be realized, offering predictions for magnetic susceptibility and entanglement dynamics at finite temperatures.

In conclusion, the work establishes that while the infinite randomness fixed point characterizes the amplitude distribution of long-range disordered chains, finite temperature introduces significant changes in sign correlations and entanglement scaling, and fundamentally alters the renormalization flow when the interaction range is sufficiently long ($\alpha < 2$).