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Rye
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I'm solving this exercise with a Heisenberg Hamiltonean in linear spin-wave theory and at some point we are asked to compute the dispersion relation at $k=0$, which leads me to finding two different stationary modes:

$$\epsilon_{+} = 2 E_0 $$

$$\epsilon_{-} = 0 $$ Where $E_0$ is a constant in the problem.

As I understand, the gapless energy mode corresponds to the Goldstone mode - which are modes that can be excited without any energy cost and are associated with slow, long-wavelength fluctuations of the order parameter (in this case, the magnetization of the ferromagnet). In the case of a ferromagnet, the Goldstone modes correspond to oscillating spin waves. When it comes to symmetry-breaking states where the order parameter (the magnetization) is not conserved, the Nambu-Goldstone modes are usually massless (thus leading us to a gapless, zero excitation energy state which can be described as a condensation of spin-wave of momentum 0). This would mean the 0 energy mode corresponds to a spontaneous symmetry breaking of the system at $k=0$

In this lens, the other mode (the "gapped" mode) should be a massive Goldstone mode. Now this is where I feel my knowledge on this topic starts to fail. From what I know, the massive Goldstone modes appear when a symmetry is explicitly broken weakly (say, for example, a small external magnetic field is added to the Heisenberg model creating an anisotropy, the existing Heiseiberg Goldstone modes that originate from the spontaneous symmetry breaking at $k=0$ become massive, attaining a finite energy, and it becomes harder to excite the system because the external potential "picks" a preferential direction for the order parameter, aka the magnetization).

What I don't understand is how this system can have both. I feel like there is gap (pun intended) in my understanding of these two definitions that is stopping me from seeing the bigger picture. Can someone help clarify these concepts and the meaning of having these two modes?

PS: In case it helps, the initial Heisenberg Hamiltonean was a Ferromagnetic model for a honeycomb lattice.

I'm solving this exercise with a Heisenberg Hamiltonean in linear spin-wave theory and at some point we are asked to compute the dispersion relation at $k=0$, which leads me to finding two different stationary modes:

$$\epsilon_{+} = 2 E_0 $$

$$\epsilon_{-} = 0 $$ Where $E_0$ is a constant in the problem.

As I understand, the gapless energy mode corresponds to the Goldstone mode - which are modes that can be excited without any energy cost and are associated with slow, long-wavelength fluctuations of the order parameter (in this case, the magnetization of the ferromagnet). In the case of a ferromagnet, the Goldstone modes correspond to oscillating spin waves. When it comes to symmetry-breaking states where the order parameter (the magnetization) is not conserved, the Nambu-Goldstone modes are usually massless (thus leading us to a gapless, zero excitation energy state which can be described as a condensation of spin-wave of momentum 0). This would mean the 0 energy mode corresponds to a spontaneous symmetry breaking of the system at $k=0$

In this lens, the other mode (the "gapped" mode) should be a massive Goldstone mode. Now this is where I feel my knowledge on this topic starts to fail. From what I know, the massive Goldstone modes appear when a symmetry is explicitly broken weakly (say, for example, a small external magnetic field is added to the Heisenberg model creating an anisotropy, the existing Heiseiberg Goldstone modes that originate from the spontaneous symmetry breaking at $k=0$ become massive, attaining a finite energy, and it becomes harder to excite the system because the external potential "picks" a preferential direction for the order parameter, aka the magnetization).

What I don't understand is how this system can have both. I feel like there is gap (pun intended) in my understanding of these two definitions that is stopping me from seeing the bigger picture. Can someone help clarify these concepts and the meaning of having these two modes?

I'm solving this exercise with a Heisenberg Hamiltonean in linear spin-wave theory and at some point we are asked to compute the dispersion relation at $k=0$, which leads me to finding two different stationary modes:

$$\epsilon_{+} = 2 E_0 $$

$$\epsilon_{-} = 0 $$ Where $E_0$ is a constant in the problem.

As I understand, the gapless energy mode corresponds to the Goldstone mode - which are modes that can be excited without any energy cost and are associated with slow, long-wavelength fluctuations of the order parameter (in this case, the magnetization of the ferromagnet). In the case of a ferromagnet, the Goldstone modes correspond to oscillating spin waves. When it comes to symmetry-breaking states where the order parameter (the magnetization) is not conserved, the Nambu-Goldstone modes are usually massless (thus leading us to a gapless, zero excitation energy state which can be described as a condensation of spin-wave of momentum 0). This would mean the 0 energy mode corresponds to a spontaneous symmetry breaking of the system at $k=0$

In this lens, the other mode (the "gapped" mode) should be a massive Goldstone mode. Now this is where I feel my knowledge on this topic starts to fail. From what I know, the massive Goldstone modes appear when a symmetry is explicitly broken weakly (say, for example, a small external magnetic field is added to the Heisenberg model creating an anisotropy, the existing Heiseiberg Goldstone modes that originate from the spontaneous symmetry breaking at $k=0$ become massive, attaining a finite energy, and it becomes harder to excite the system because the external potential "picks" a preferential direction for the order parameter, aka the magnetization).

What I don't understand is how this system can have both. I feel like there is gap (pun intended) in my understanding of these two definitions that is stopping me from seeing the bigger picture. Can someone help clarify these concepts and the meaning of having these two modes?

PS: In case it helps, the initial Heisenberg Hamiltonean was a Ferromagnetic model for a honeycomb lattice.

Source Link
Rye
Rye
  • 548
  • 4
  • 19

Explanation of massive Goldstone modes

I'm solving this exercise with a Heisenberg Hamiltonean in linear spin-wave theory and at some point we are asked to compute the dispersion relation at $k=0$, which leads me to finding two different stationary modes:

$$\epsilon_{+} = 2 E_0 $$

$$\epsilon_{-} = 0 $$ Where $E_0$ is a constant in the problem.

As I understand, the gapless energy mode corresponds to the Goldstone mode - which are modes that can be excited without any energy cost and are associated with slow, long-wavelength fluctuations of the order parameter (in this case, the magnetization of the ferromagnet). In the case of a ferromagnet, the Goldstone modes correspond to oscillating spin waves. When it comes to symmetry-breaking states where the order parameter (the magnetization) is not conserved, the Nambu-Goldstone modes are usually massless (thus leading us to a gapless, zero excitation energy state which can be described as a condensation of spin-wave of momentum 0). This would mean the 0 energy mode corresponds to a spontaneous symmetry breaking of the system at $k=0$

In this lens, the other mode (the "gapped" mode) should be a massive Goldstone mode. Now this is where I feel my knowledge on this topic starts to fail. From what I know, the massive Goldstone modes appear when a symmetry is explicitly broken weakly (say, for example, a small external magnetic field is added to the Heisenberg model creating an anisotropy, the existing Heiseiberg Goldstone modes that originate from the spontaneous symmetry breaking at $k=0$ become massive, attaining a finite energy, and it becomes harder to excite the system because the external potential "picks" a preferential direction for the order parameter, aka the magnetization).

What I don't understand is how this system can have both. I feel like there is gap (pun intended) in my understanding of these two definitions that is stopping me from seeing the bigger picture. Can someone help clarify these concepts and the meaning of having these two modes?